Calculus Examples

$f (x) = 6 x^{5} - 75 x^{4} + 300 x^{3} - 375 x^{2} - 1$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $6 x^{5} - 75 x^{4} + 300 x^{3} - 375 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.2

Evaluate $\frac{d}{d x} [6 x^{5}]$ .

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Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{5}$ with respect to $x$ is $6 \frac{d}{d x} [x^{5}]$ .

$6 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$6 (5 x^{4}) + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.2.3

Multiply $5$ by $6$ .

$30 x^{4} + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.3

Evaluate $\frac{d}{d x} [- 75 x^{4}]$ .

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Step 1.3.1

Since $- 75$ is constant with respect to $x$ , the derivative of $- 75 x^{4}$ with respect to $x$ is $- 75 \frac{d}{d x} [x^{4}]$ .

$30 x^{4} - 75 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$30 x^{4} - 75 (4 x^{3}) + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.3.3

Multiply $4$ by $- 75$ .

$30 x^{4} - 300 x^{3} + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.4

Evaluate $\frac{d}{d x} [300 x^{3}]$ .

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Step 1.4.1

Since $300$ is constant with respect to $x$ , the derivative of $300 x^{3}$ with respect to $x$ is $300 \frac{d}{d x} [x^{3}]$ .

$30 x^{4} - 300 x^{3} + 300 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$30 x^{4} - 300 x^{3} + 300 (3 x^{2}) + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.4.3

Multiply $3$ by $300$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.5

Evaluate $\frac{d}{d x} [- 375 x^{2}]$ .

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Step 1.5.1

Since $- 375$ is constant with respect to $x$ , the derivative of $- 375 x^{2}$ with respect to $x$ is $- 375 \frac{d}{d x} [x^{2}]$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 375 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$

Step 1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 375 (2 x) + \frac{d}{d x} [- 1]$

Step 1.5.3

Multiply $2$ by $- 375$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x + \frac{d}{d x} [- 1]$

Step 1.6

Differentiate using the Constant Rule.

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Step 1.6.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x + 0$

Step 1.6.2

Add $30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$ and $0$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$ with respect to $x$ is $\frac{d}{d x} [30 x^{4}] + \frac{d}{d x} [- 300 x^{3}] + \frac{d}{d x} [900 x^{2}] + \frac{d}{d x} [- 750 x]$ .

$f'' (x) = \frac{d}{d x} (30 x^{4}) + \frac{d}{d x} (- 300 x^{3}) + \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.2

Evaluate $\frac{d}{d x} [30 x^{4}]$ .

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Step 2.2.1

Since $30$ is constant with respect to $x$ , the derivative of $30 x^{4}$ with respect to $x$ is $30 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 30 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 300 x^{3}) + \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = 30 (4 x^{3}) + \frac{d}{d x} (- 300 x^{3}) + \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.2.3

Multiply $4$ by $30$ .

$f'' (x) = 120 x^{3} + \frac{d}{d x} (- 300 x^{3}) + \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.3

Evaluate $\frac{d}{d x} [- 300 x^{3}]$ .

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Step 2.3.1

Since $- 300$ is constant with respect to $x$ , the derivative of $- 300 x^{3}$ with respect to $x$ is $- 300 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 120 x^{3} - 300 \frac{d}{d x} x^{3} + \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 120 x^{3} - 300 (3 x^{2}) + \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.3.3

Multiply $3$ by $- 300$ .

$f'' (x) = 120 x^{3} - 900 x^{2} + \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.4

Evaluate $\frac{d}{d x} [900 x^{2}]$ .

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Step 2.4.1

Since $900$ is constant with respect to $x$ , the derivative of $900 x^{2}$ with respect to $x$ is $900 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 120 x^{3} - 900 x^{2} + 900 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 750 x)$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 120 x^{3} - 900 x^{2} + 900 (2 x) + \frac{d}{d x} (- 750 x)$

Step 2.4.3

Multiply $2$ by $900$ .

$f'' (x) = 120 x^{3} - 900 x^{2} + 1800 x + \frac{d}{d x} (- 750 x)$

Step 2.5

Evaluate $\frac{d}{d x} [- 750 x]$ .

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Step 2.5.1

Since $- 750$ is constant with respect to $x$ , the derivative of $- 750 x$ with respect to $x$ is $- 750 \frac{d}{d x} [x]$ .

$f'' (x) = 120 x^{3} - 900 x^{2} + 1800 x - 750 \frac{d}{d x} x$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 120 x^{3} - 900 x^{2} + 1800 x - 750 \cdot 1$

Step 2.5.3

Multiply $- 750$ by $1$ .

$f'' (x) = 120 x^{3} - 900 x^{2} + 1800 x - 750$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.2

Evaluate $\frac{d}{d x} [6 x^{5}]$ .

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Step 4.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{5}$ with respect to $x$ is $6 \frac{d}{d x} [x^{5}]$ .

$6 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$6 (5 x^{4}) + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.2.3

Multiply $5$ by $6$ .

$30 x^{4} + \frac{d}{d x} [- 75 x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 75 x^{4}]$ .

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Step 4.1.3.1

Since $- 75$ is constant with respect to $x$ , the derivative of $- 75 x^{4}$ with respect to $x$ is $- 75 \frac{d}{d x} [x^{4}]$ .

$30 x^{4} - 75 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$30 x^{4} - 75 (4 x^{3}) + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.3.3

Multiply $4$ by $- 75$ .

$30 x^{4} - 300 x^{3} + \frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.4

Evaluate $\frac{d}{d x} [300 x^{3}]$ .

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Step 4.1.4.1

Since $300$ is constant with respect to $x$ , the derivative of $300 x^{3}$ with respect to $x$ is $300 \frac{d}{d x} [x^{3}]$ .

$30 x^{4} - 300 x^{3} + 300 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$30 x^{4} - 300 x^{3} + 300 (3 x^{2}) + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.4.3

Multiply $3$ by $300$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} + \frac{d}{d x} [- 375 x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.5

Evaluate $\frac{d}{d x} [- 375 x^{2}]$ .

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Step 4.1.5.1

Since $- 375$ is constant with respect to $x$ , the derivative of $- 375 x^{2}$ with respect to $x$ is $- 375 \frac{d}{d x} [x^{2}]$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 375 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$

Step 4.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 375 (2 x) + \frac{d}{d x} [- 1]$

Step 4.1.5.3

Multiply $2$ by $- 375$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x + \frac{d}{d x} [- 1]$

Step 4.1.6

Differentiate using the Constant Rule.

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Step 4.1.6.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x + 0$

Step 4.1.6.2

Add $30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$ and $0$ .

$f' (x) = 30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$

Step 5

Set the first derivative equal to $0$ then solve the equation $30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $30 x$ out of $30 x^{4} - 300 x^{3} + 900 x^{2} - 750 x$ .

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Step 5.2.1.1

Factor $30 x$ out of $30 x^{4}$ .

$30 x (x^{3}) - 300 x^{3} + 900 x^{2} - 750 x = 0$

Step 5.2.1.2

Factor $30 x$ out of $- 300 x^{3}$ .

$30 x (x^{3}) + 30 x (- 10 x^{2}) + 900 x^{2} - 750 x = 0$

Step 5.2.1.3

Factor $30 x$ out of $900 x^{2}$ .

$30 x (x^{3}) + 30 x (- 10 x^{2}) + 30 x (30 x) - 750 x = 0$

Step 5.2.1.4

Factor $30 x$ out of $- 750 x$ .

$30 x (x^{3}) + 30 x (- 10 x^{2}) + 30 x (30 x) + 30 x (- 25) = 0$

Step 5.2.1.5

Factor $30 x$ out of $30 x (x^{3}) + 30 x (- 10 x^{2})$ .

$30 x (x^{3} - 10 x^{2}) + 30 x (30 x) + 30 x (- 25) = 0$

Step 5.2.1.6

Factor $30 x$ out of $30 x (x^{3} - 10 x^{2}) + 30 x (30 x)$ .

$30 x (x^{3} - 10 x^{2} + 30 x) + 30 x (- 25) = 0$

Step 5.2.1.7

Factor $30 x$ out of $30 x (x^{3} - 10 x^{2} + 30 x) + 30 x (- 25)$ .

$30 x (x^{3} - 10 x^{2} + 30 x - 25) = 0$

Step 5.2.2

Factor.

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Step 5.2.2.1

Factor $x^{3} - 10 x^{2} + 30 x - 25$ using the rational roots test.

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Step 5.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 25, \pm 5$

$q = \pm 1$

Step 5.2.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 25, \pm 5$

Step 5.2.2.1.3

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

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Step 5.2.2.1.3.1

Substitute $5$ into the polynomial.

$5^{3} - 10 \cdot 5^{2} + 30 \cdot 5 - 25$

Step 5.2.2.1.3.2

Raise $5$ to the power of $3$ .

$125 - 10 \cdot 5^{2} + 30 \cdot 5 - 25$

Step 5.2.2.1.3.3

Raise $5$ to the power of $2$ .

$125 - 10 \cdot 25 + 30 \cdot 5 - 25$

Step 5.2.2.1.3.4

Multiply $- 10$ by $25$ .

$125 - 250 + 30 \cdot 5 - 25$

Step 5.2.2.1.3.5

Subtract $250$ from $125$ .

$- 125 + 30 \cdot 5 - 25$

Step 5.2.2.1.3.6

Multiply $30$ by $5$ .

$- 125 + 150 - 25$

Step 5.2.2.1.3.7

Add $- 125$ and $150$ .

$25 - 25$

Step 5.2.2.1.3.8

Subtract $25$ from $25$ .

$0$

Step 5.2.2.1.4

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 10 x^{2} + 30 x - 25}{x - 5}$

Step 5.2.2.1.5

Divide $x^{3} - 10 x^{2} + 30 x - 25$ by $x - 5$ .

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Step 5.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$5$

$x^{3}$

$10 x^{2}$

$30 x$

$25$

Step 5.2.2.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$

Step 5.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			+	$x^{3}$	-	$5 x^{2}$

Step 5.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 5 x^{2}$

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$

Step 5.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$

Step 5.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$

Step 5.2.2.1.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$

Step 5.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					-	$5 x^{2}$	+	$25 x$

Step 5.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} + 25 x$

				$x^{2}$	-	$5 x$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					+	$5 x^{2}$	-	$25 x$

Step 5.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$5 x$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					+	$5 x^{2}$	-	$25 x$
							+	$5 x$

Step 5.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$5 x$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					+	$5 x^{2}$	-	$25 x$
							+	$5 x$	-	$25$

Step 5.2.2.1.5.12

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$	+	$5$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					+	$5 x^{2}$	-	$25 x$
							+	$5 x$	-	$25$

Step 5.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	+	$5$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					+	$5 x^{2}$	-	$25 x$
							+	$5 x$	-	$25$
							+	$5 x$	-	$25$

Step 5.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $5 x - 25$

				$x^{2}$	-	$5 x$	+	$5$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					+	$5 x^{2}$	-	$25 x$
							+	$5 x$	-	$25$
							-	$5 x$	+	$25$

Step 5.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$5 x$	+	$5$
$x$	-	$5$		$x^{3}$	-	$10 x^{2}$	+	$30 x$	-	$25$
			-	$x^{3}$	+	$5 x^{2}$
					-	$5 x^{2}$	+	$30 x$
					+	$5 x^{2}$	-	$25 x$
							+	$5 x$	-	$25$
							-	$5 x$	+	$25$
										$0$

Step 5.2.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 5 x + 5$

Step 5.2.2.1.6

Write $x^{3} - 10 x^{2} + 30 x - 25$ as a set of factors.

$30 x ((x - 5) (x^{2} - 5 x + 5)) = 0$

Step 5.2.2.2

Remove unnecessary parentheses.

$30 x (x - 5) (x^{2} - 5 x + 5) = 0$

Step 5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 5 = 0$

$x^{2} - 5 x + 5 = 0$

Step 5.4

Set $x$ equal to $0$ .

$x = 0$

Step 5.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 5.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 5.6

Set $x^{2} - 5 x + 5$ equal to $0$ and solve for $x$ .

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Step 5.6.1

Set $x^{2} - 5 x + 5$ equal to $0$ .

$x^{2} - 5 x + 5 = 0$

Step 5.6.2

Solve $x^{2} - 5 x + 5 = 0$ for $x$ .

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Step 5.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.6.2.2

Substitute the values $a = 1$ , $b = - 5$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Step 5.6.2.3

Simplify.

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Step 5.6.2.3.1

Simplify the numerator.

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Step 5.6.2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 5.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 5.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5}}{2 \cdot 1}$

Step 5.6.2.3.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20}}{2 \cdot 1}$

Step 5.6.2.3.1.3

Subtract $20$ from $25$ .

$x = \frac{5 \pm \sqrt{5}}{2 \cdot 1}$

Step 5.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{5}}{2}$

Step 5.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.6.2.4.1

Simplify the numerator.

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Step 5.6.2.4.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 5.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 5.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5}}{2 \cdot 1}$

Step 5.6.2.4.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20}}{2 \cdot 1}$

Step 5.6.2.4.1.3

Subtract $20$ from $25$ .

$x = \frac{5 \pm \sqrt{5}}{2 \cdot 1}$

Step 5.6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{5}}{2}$

Step 5.6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{5}}{2}$

Step 5.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.6.2.5.1

Simplify the numerator.

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Step 5.6.2.5.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 5.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 5.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5}}{2 \cdot 1}$

Step 5.6.2.5.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20}}{2 \cdot 1}$

Step 5.6.2.5.1.3

Subtract $20$ from $25$ .

$x = \frac{5 \pm \sqrt{5}}{2 \cdot 1}$

Step 5.6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{5}}{2}$

Step 5.6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{5}}{2}$

Step 5.6.2.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{5}}{2}, \frac{5 - \sqrt{5}}{2}$

Step 5.7

The final solution is all the values that make $30 x (x - 5) (x^{2} - 5 x + 5) = 0$ true.

$x = 0, 5, \frac{5 + \sqrt{5}}{2}, \frac{5 - \sqrt{5}}{2}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0, 5, \frac{5 + \sqrt{5}}{2}, \frac{5 - \sqrt{5}}{2}$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$120 {(0)}^{3} - 900 {(0)}^{2} + 1800 (0) - 750$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Raising $0$ to any positive power yields $0$ .

$120 \cdot 0 - 900 {(0)}^{2} + 1800 (0) - 750$

Step 9.1.2

Multiply $120$ by $0$ .

$0 - 900 {(0)}^{2} + 1800 (0) - 750$

Step 9.1.3

Raising $0$ to any positive power yields $0$ .

$0 - 900 \cdot 0 + 1800 (0) - 750$

Step 9.1.4

Multiply $- 900$ by $0$ .

$0 + 0 + 1800 (0) - 750$

Step 9.1.5

Multiply $1800$ by $0$ .

$0 + 0 + 0 - 750$

Step 9.2

Simplify by adding and subtracting.

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Step 9.2.1

Add $0$ and $0$ .

$0 + 0 - 750$

Step 9.2.2

Add $0$ and $0$ .

$0 - 750$

Step 9.2.3

Subtract $750$ from $0$ .

$- 750$

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 6 {(0)}^{5} - 75 {(0)}^{4} + 300 {(0)}^{3} - 375 {(0)}^{2} - 1$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 6 \cdot 0 - 75 {(0)}^{4} + 300 {(0)}^{3} - 375 {(0)}^{2} - 1$

Step 11.2.1.2

Multiply $6$ by $0$ .

$f (0) = 0 - 75 {(0)}^{4} + 300 {(0)}^{3} - 375 {(0)}^{2} - 1$

Step 11.2.1.3

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 75 \cdot 0 + 300 {(0)}^{3} - 375 {(0)}^{2} - 1$

Step 11.2.1.4

Multiply $- 75$ by $0$ .

$f (0) = 0 + 0 + 300 {(0)}^{3} - 375 {(0)}^{2} - 1$

Step 11.2.1.5

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 + 300 \cdot 0 - 375 {(0)}^{2} - 1$

Step 11.2.1.6

Multiply $300$ by $0$ .

$f (0) = 0 + 0 + 0 - 375 {(0)}^{2} - 1$

Step 11.2.1.7

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 + 0 - 375 \cdot 0 - 1$

Step 11.2.1.8

Multiply $- 375$ by $0$ .

$f (0) = 0 + 0 + 0 + 0 - 1$

Step 11.2.2

Simplify by adding and subtracting.

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Step 11.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 0 - 1$

Step 11.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 0 - 1$

Step 11.2.2.3

Add $0$ and $0$ .

$f (0) = 0 - 1$

Step 11.2.2.4

Subtract $1$ from $0$ .

$f (0) = - 1$

Step 11.2.3

The final answer is $- 1$ .

$y = - 1$

Step 12

Evaluate the second derivative at $x = 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$120 {(5)}^{3} - 900 {(5)}^{2} + 1800 (5) - 750$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Raise $5$ to the power of $3$ .

$120 \cdot 125 - 900 {(5)}^{2} + 1800 (5) - 750$

Step 13.1.2

Multiply $120$ by $125$ .

$15000 - 900 {(5)}^{2} + 1800 (5) - 750$

Step 13.1.3

Raise $5$ to the power of $2$ .

$15000 - 900 \cdot 25 + 1800 (5) - 750$

Step 13.1.4

Multiply $- 900$ by $25$ .

$15000 - 22500 + 1800 (5) - 750$

Step 13.1.5

Multiply $1800$ by $5$ .

$15000 - 22500 + 9000 - 750$

Step 13.2

Simplify by adding and subtracting.

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Step 13.2.1

Subtract $22500$ from $15000$ .

$- 7500 + 9000 - 750$

Step 13.2.2

Add $- 7500$ and $9000$ .

$1500 - 750$

Step 13.2.3

Subtract $750$ from $1500$ .

$750$

Step 14

$x = 5$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 5$ is a local minimum

Step 15

Find the y-value when $x = 5$ .

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Step 15.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = 6 {(5)}^{5} - 75 {(5)}^{4} + 300 {(5)}^{3} - 375 {(5)}^{2} - 1$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Raise $5$ to the power of $5$ .

$f (5) = 6 \cdot 3125 - 75 {(5)}^{4} + 300 {(5)}^{3} - 375 {(5)}^{2} - 1$

Step 15.2.1.2

Multiply $6$ by $3125$ .

$f (5) = 18750 - 75 {(5)}^{4} + 300 {(5)}^{3} - 375 {(5)}^{2} - 1$

Step 15.2.1.3

Raise $5$ to the power of $4$ .

$f (5) = 18750 - 75 \cdot 625 + 300 {(5)}^{3} - 375 {(5)}^{2} - 1$

Step 15.2.1.4

Multiply $- 75$ by $625$ .

$f (5) = 18750 - 46875 + 300 {(5)}^{3} - 375 {(5)}^{2} - 1$

Step 15.2.1.5

Raise $5$ to the power of $3$ .

$f (5) = 18750 - 46875 + 300 \cdot 125 - 375 {(5)}^{2} - 1$

Step 15.2.1.6

Multiply $300$ by $125$ .

$f (5) = 18750 - 46875 + 37500 - 375 {(5)}^{2} - 1$

Step 15.2.1.7

Raise $5$ to the power of $2$ .

$f (5) = 18750 - 46875 + 37500 - 375 \cdot 25 - 1$

Step 15.2.1.8

Multiply $- 375$ by $25$ .

$f (5) = 18750 - 46875 + 37500 - 9375 - 1$

Step 15.2.2

Simplify by adding and subtracting.

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Step 15.2.2.1

Subtract $46875$ from $18750$ .

$f (5) = - 28125 + 37500 - 9375 - 1$

Step 15.2.2.2

Add $- 28125$ and $37500$ .

$f (5) = 9375 - 9375 - 1$

Step 15.2.2.3

Subtract $9375$ from $9375$ .

$f (5) = 0 - 1$

Step 15.2.2.4

Subtract $1$ from $0$ .

$f (5) = - 1$

Step 15.2.3

The final answer is $- 1$ .

$y = - 1$

Step 16

Evaluate the second derivative at $x = \frac{5 + \sqrt{5}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$120 {(\frac{5 + \sqrt{5}}{2})}^{3} - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

Apply the product rule to $\frac{5 + \sqrt{5}}{2}$ .

$120 \frac{{(5 + \sqrt{5})}^{3}}{2^{3}} - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.2

Raise $2$ to the power of $3$ .

$120 \frac{{(5 + \sqrt{5})}^{3}}{8} - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.3

Cancel the common factor of $8$ .

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Step 17.1.3.1

Factor $8$ out of $120$ .

$8 (15) \frac{{(5 + \sqrt{5})}^{3}}{8} - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.3.2

Cancel the common factor.

$8 \cdot 15 \frac{{(5 + \sqrt{5})}^{3}}{8} - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.3.3

Rewrite the expression.

$15 {(5 + \sqrt{5})}^{3} - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.4

Use the Binomial Theorem.

$15 (5^{3} + 3 \cdot 5^{2} \sqrt{5} + 3 \cdot 5 {\sqrt{5}}^{2} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5

Simplify each term.

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Step 17.1.5.1

Raise $5$ to the power of $3$ .

$15 (125 + 3 \cdot 5^{2} \sqrt{5} + 3 \cdot 5 {\sqrt{5}}^{2} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.2

Raise $5$ to the power of $2$ .

$15 (125 + 3 \cdot 25 \sqrt{5} + 3 \cdot 5 {\sqrt{5}}^{2} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.3

Multiply $3$ by $25$ .

$15 (125 + 75 \sqrt{5} + 3 \cdot 5 {\sqrt{5}}^{2} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.4

Multiply $3$ by $5$ .

$15 (125 + 75 \sqrt{5} + 15 {\sqrt{5}}^{2} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 17.1.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$15 (125 + 75 \sqrt{5} + 15 {(5^{\frac{1}{2}})}^{2} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$15 (125 + 75 \sqrt{5} + 15 \cdot 5^{\frac{1}{2} \cdot 2} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.5.3

Combine $\frac{1}{2}$ and $2$ .

$15 (125 + 75 \sqrt{5} + 15 \cdot 5^{\frac{2}{2}} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.5.4

Cancel the common factor of $2$ .

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Step 17.1.5.5.4.1

Cancel the common factor.

$15 (125 + 75 \sqrt{5} + 15 \cdot 5^{\frac{2}{2}} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.5.4.2

Rewrite the expression.

$15 (125 + 75 \sqrt{5} + 15 \cdot 5^{1} + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.5.5

Evaluate the exponent.

$15 (125 + 75 \sqrt{5} + 15 \cdot 5 + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.6

Multiply $15$ by $5$ .

$15 (125 + 75 \sqrt{5} + 75 + {\sqrt{5}}^{3}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.7

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$15 (125 + 75 \sqrt{5} + 75 + \sqrt{5^{3}}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.8

Raise $5$ to the power of $3$ .

$15 (125 + 75 \sqrt{5} + 75 + \sqrt{125}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.9

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 17.1.5.9.1

Factor $25$ out of $125$ .

$15 (125 + 75 \sqrt{5} + 75 + \sqrt{25 (5)}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.9.2

Rewrite $25$ as $5^{2}$ .

$15 (125 + 75 \sqrt{5} + 75 + \sqrt{5^{2} \cdot 5}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.5.10

Pull terms out from under the radical.

$15 (125 + 75 \sqrt{5} + 75 + 5 \sqrt{5}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.6

Add $125$ and $75$ .

$15 (200 + 75 \sqrt{5} + 5 \sqrt{5}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.7

Add $75 \sqrt{5}$ and $5 \sqrt{5}$ .

$15 (200 + 80 \sqrt{5}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.8

Apply the distributive property.

$15 \cdot 200 + 15 (80 \sqrt{5}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.9

Multiply $15$ by $200$ .

$3000 + 15 (80 \sqrt{5}) - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.10

Multiply $80$ by $15$ .

$3000 + 1200 \sqrt{5} - 900 {(\frac{5 + \sqrt{5}}{2})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.11

Apply the product rule to $\frac{5 + \sqrt{5}}{2}$ .

$3000 + 1200 \sqrt{5} - 900 \frac{{(5 + \sqrt{5})}^{2}}{2^{2}} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.12

Raise $2$ to the power of $2$ .

$3000 + 1200 \sqrt{5} - 900 \frac{{(5 + \sqrt{5})}^{2}}{4} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.13

Cancel the common factor of $4$ .

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Step 17.1.13.1

Factor $4$ out of $- 900$ .

$3000 + 1200 \sqrt{5} + 4 (- 225) \frac{{(5 + \sqrt{5})}^{2}}{4} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.13.2

Cancel the common factor.

$3000 + 1200 \sqrt{5} + 4 \cdot - 225 \frac{{(5 + \sqrt{5})}^{2}}{4} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.13.3

Rewrite the expression.

$3000 + 1200 \sqrt{5} - 225 {(5 + \sqrt{5})}^{2} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.14

Rewrite ${(5 + \sqrt{5})}^{2}$ as $(5 + \sqrt{5}) (5 + \sqrt{5})$ .

$3000 + 1200 \sqrt{5} - 225 ((5 + \sqrt{5}) (5 + \sqrt{5})) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.15

Expand $(5 + \sqrt{5}) (5 + \sqrt{5})$ using the FOIL Method.

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Step 17.1.15.1

Apply the distributive property.

$3000 + 1200 \sqrt{5} - 225 (5 (5 + \sqrt{5}) + \sqrt{5} (5 + \sqrt{5})) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.15.2

Apply the distributive property.

$3000 + 1200 \sqrt{5} - 225 (5 \cdot 5 + 5 \sqrt{5} + \sqrt{5} (5 + \sqrt{5})) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.15.3

Apply the distributive property.

$3000 + 1200 \sqrt{5} - 225 (5 \cdot 5 + 5 \sqrt{5} + \sqrt{5} \cdot 5 + \sqrt{5} \sqrt{5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16

Simplify and combine like terms.

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Step 17.1.16.1

Simplify each term.

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Step 17.1.16.1.1

Multiply $5$ by $5$ .

$3000 + 1200 \sqrt{5} - 225 (25 + 5 \sqrt{5} + \sqrt{5} \cdot 5 + \sqrt{5} \sqrt{5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16.1.2

Move $5$ to the left of $\sqrt{5}$ .

$3000 + 1200 \sqrt{5} - 225 (25 + 5 \sqrt{5} + 5 \cdot \sqrt{5} + \sqrt{5} \sqrt{5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16.1.3

Combine using the product rule for radicals.

$3000 + 1200 \sqrt{5} - 225 (25 + 5 \sqrt{5} + 5 \sqrt{5} + \sqrt{5 \cdot 5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16.1.4

Multiply $5$ by $5$ .

$3000 + 1200 \sqrt{5} - 225 (25 + 5 \sqrt{5} + 5 \sqrt{5} + \sqrt{25}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16.1.5

Rewrite $25$ as $5^{2}$ .

$3000 + 1200 \sqrt{5} - 225 (25 + 5 \sqrt{5} + 5 \sqrt{5} + \sqrt{5^{2}}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16.1.6

Pull terms out from under the radical, assuming positive real numbers.

$3000 + 1200 \sqrt{5} - 225 (25 + 5 \sqrt{5} + 5 \sqrt{5} + 5) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16.2

Add $25$ and $5$ .

$3000 + 1200 \sqrt{5} - 225 (30 + 5 \sqrt{5} + 5 \sqrt{5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.16.3

Add $5 \sqrt{5}$ and $5 \sqrt{5}$ .

$3000 + 1200 \sqrt{5} - 225 (30 + 10 \sqrt{5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.17

Apply the distributive property.

$3000 + 1200 \sqrt{5} - 225 \cdot 30 - 225 (10 \sqrt{5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.18

Multiply $- 225$ by $30$ .

$3000 + 1200 \sqrt{5} - 6750 - 225 (10 \sqrt{5}) + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.19

Multiply $10$ by $- 225$ .

$3000 + 1200 \sqrt{5} - 6750 - 2250 \sqrt{5} + 1800 (\frac{5 + \sqrt{5}}{2}) - 750$

Step 17.1.20

Cancel the common factor of $2$ .

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Step 17.1.20.1

Factor $2$ out of $1800$ .

$3000 + 1200 \sqrt{5} - 6750 - 2250 \sqrt{5} + 2 (900) \frac{5 + \sqrt{5}}{2} - 750$

Step 17.1.20.2

Cancel the common factor.

$3000 + 1200 \sqrt{5} - 6750 - 2250 \sqrt{5} + 2 \cdot 900 \frac{5 + \sqrt{5}}{2} - 750$

Step 17.1.20.3

Rewrite the expression.

$3000 + 1200 \sqrt{5} - 6750 - 2250 \sqrt{5} + 900 (5 + \sqrt{5}) - 750$

Step 17.1.21

Apply the distributive property.

$3000 + 1200 \sqrt{5} - 6750 - 2250 \sqrt{5} + 900 \cdot 5 + 900 \sqrt{5} - 750$

Step 17.1.22

Multiply $900$ by $5$ .

$3000 + 1200 \sqrt{5} - 6750 - 2250 \sqrt{5} + 4500 + 900 \sqrt{5} - 750$

Step 17.2

Simplify by adding terms.

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Step 17.2.1

Subtract $6750$ from $3000$ .

$- 3750 + 1200 \sqrt{5} - 2250 \sqrt{5} + 4500 + 900 \sqrt{5} - 750$

Step 17.2.2

Simplify by adding and subtracting.

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Step 17.2.2.1

Add $- 3750$ and $4500$ .

$750 + 1200 \sqrt{5} - 2250 \sqrt{5} + 900 \sqrt{5} - 750$

Step 17.2.2.2

Subtract $750$ from $750$ .

$0 + 1200 \sqrt{5} - 2250 \sqrt{5} + 900 \sqrt{5}$

Step 17.2.2.3

Add $0$ and $1200 \sqrt{5}$ .

$1200 \sqrt{5} - 2250 \sqrt{5} + 900 \sqrt{5}$

Step 17.2.3

Subtract $2250 \sqrt{5}$ from $1200 \sqrt{5}$ .

$- 1050 \sqrt{5} + 900 \sqrt{5}$

Step 17.2.4

Add $- 1050 \sqrt{5}$ and $900 \sqrt{5}$ .

$- 150 \sqrt{5}$

Step 18

$x = \frac{5 + \sqrt{5}}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{5 + \sqrt{5}}{2}$ is a local maximum

Step 19

Find the y-value when $x = \frac{5 + \sqrt{5}}{2}$ .

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Step 19.1

Replace the variable $x$ with $\frac{5 + \sqrt{5}}{2}$ in the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 6 {(\frac{5 + \sqrt{5}}{2})}^{5} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Apply the product rule to $\frac{5 + \sqrt{5}}{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 6 (\frac{{(5 + \sqrt{5})}^{5}}{2^{5}}) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.2

Raise $2$ to the power of $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 6 (\frac{{(5 + \sqrt{5})}^{5}}{32}) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.3

Cancel the common factor of $2$ .

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Step 19.2.1.3.1

Factor $2$ out of $6$ .

$f (\frac{5 + \sqrt{5}}{2}) = 2 (3) (\frac{{(5 + \sqrt{5})}^{5}}{32}) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.3.2

Factor $2$ out of $32$ .

$f (\frac{5 + \sqrt{5}}{2}) = 2 \cdot (3 (\frac{{(5 + \sqrt{5})}^{5}}{2 \cdot 16})) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.3.3

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = 2 \cdot (3 (\frac{{(5 + \sqrt{5})}^{5}}{2 \cdot 16})) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.3.4

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 3 (\frac{{(5 + \sqrt{5})}^{5}}{16}) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.4

Combine $3$ and $\frac{{(5 + \sqrt{5})}^{5}}{16}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 {(5 + \sqrt{5})}^{5}}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.5

Use the Binomial Theorem.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (5^{5} + 5 \cdot (5^{4} \sqrt{5}) + 10 \cdot (5^{3} {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6

Simplify each term.

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Step 19.2.1.6.1

Raise $5$ to the power of $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 5 \cdot (5^{4} \sqrt{5}) + 10 \cdot (5^{3} {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.2

Multiply $5$ by $5^{4}$ by adding the exponents.

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Step 19.2.1.6.2.1

Multiply $5$ by $5^{4}$ .

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Step 19.2.1.6.2.1.1

Raise $5$ to the power of $1$ .

Step 19.2.1.6.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 5^{1 + 4} \sqrt{5} + 10 \cdot (5^{3} {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.2.2

Add $1$ and $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 5^{5} \sqrt{5} + 10 \cdot (5^{3} {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.3

Raise $5$ to the power of $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 10 \cdot (5^{3} {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.4

Raise $5$ to the power of $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 10 \cdot (125 {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.5

Multiply $10$ by $125$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 1250 {\sqrt{5}}^{2} + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 19.2.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 1250 {(5^{\frac{1}{2}})}^{2} + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 1250 \cdot 5^{\frac{1}{2} \cdot 2} + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 1250 \cdot 5^{\frac{2}{2}} + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.6.4

Cancel the common factor of $2$ .

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Step 19.2.1.6.6.4.1

Cancel the common factor.

Step 19.2.1.6.6.4.2

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 1250 \cdot 5 + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.6.5

Evaluate the exponent.

Step 19.2.1.6.7

Multiply $1250$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 10 \cdot (5^{2} {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.8

Raise $5$ to the power of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 10 \cdot (25 {\sqrt{5}}^{3}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.9

Multiply $10$ by $25$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 250 {\sqrt{5}}^{3} + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.10

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 250 \sqrt{5^{3}} + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.11

Raise $5$ to the power of $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 250 \sqrt{125} + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.12

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 19.2.1.6.12.1

Factor $25$ out of $125$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 250 \sqrt{25 (5)} + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.12.2

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 250 \sqrt{5^{2} \cdot 5} + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.13

Pull terms out from under the radical.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 250 (5 \sqrt{5}) + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.14

Multiply $5$ by $250$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 5 \cdot (5 {\sqrt{5}}^{4}) + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.15

Multiply $5$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 {\sqrt{5}}^{4} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 19.2.1.6.16.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 {(5^{\frac{1}{2}})}^{4} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 5^{\frac{1}{2} \cdot 4} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16.3

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 5^{\frac{4}{2}} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16.4

Cancel the common factor of $4$ and $2$ .

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Step 19.2.1.6.16.4.1

Factor $2$ out of $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 5^{\frac{2 \cdot 2}{2}} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16.4.2

Cancel the common factors.

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Step 19.2.1.6.16.4.2.1

Factor $2$ out of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 5^{\frac{2 \cdot 2}{2 (1)}} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16.4.2.2

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 5^{\frac{2 \cdot 2}{2 \cdot 1}} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16.4.2.3

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 5^{\frac{2}{1}} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.16.4.2.4

Divide $2$ by $1$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 5^{2} + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.17

Raise $5$ to the power of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 25 \cdot 25 + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.18

Multiply $25$ by $25$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 625 + {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.19

Rewrite ${\sqrt{5}}^{5}$ as $\sqrt{5^{5}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 625 + \sqrt{5^{5}})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.20

Raise $5$ to the power of $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 625 + \sqrt{3125})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.21

Rewrite $3125$ as $25^{2} \cdot 5$ .

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Step 19.2.1.6.21.1

Factor $625$ out of $3125$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 625 + \sqrt{625 (5)})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.21.2

Rewrite $625$ as $25^{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 625 + \sqrt{25^{2} \cdot 5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.6.22

Pull terms out from under the radical.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (3125 + 3125 \sqrt{5} + 6250 + 1250 \sqrt{5} + 625 + 25 \sqrt{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.7

Add $3125$ and $6250$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (9375 + 3125 \sqrt{5} + 1250 \sqrt{5} + 625 + 25 \sqrt{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.8

Add $9375$ and $625$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (10000 + 3125 \sqrt{5} + 1250 \sqrt{5} + 25 \sqrt{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.9

Add $3125 \sqrt{5}$ and $1250 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (10000 + 4375 \sqrt{5} + 25 \sqrt{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.10

Add $4375 \sqrt{5}$ and $25 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (10000 + 4400 \sqrt{5})}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.11

Cancel the common factor of $10000 + 4400 \sqrt{5}$ and $16$ .

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Step 19.2.1.11.1

Factor $16$ out of $3 (10000 + 4400 \sqrt{5})$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{16 (3 (625 + 275 \sqrt{5}))}{16} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.11.2

Cancel the common factors.

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Step 19.2.1.11.2.1

Factor $16$ out of $16$ .

$f (\frac{5 + \sqrt{5}}{2}) = \frac{16 (3 (625 + 275 \sqrt{5}))}{16 (1)} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.11.2.2

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{16 (3 (625 + 275 \sqrt{5}))}{16 \cdot 1} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.11.2.3

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = \frac{3 (625 + 275 \sqrt{5})}{1} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.11.2.4

Divide $3 (625 + 275 \sqrt{5})$ by $1$ .

$f (\frac{5 + \sqrt{5}}{2}) = 3 (625 + 275 \sqrt{5}) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.12

Apply the distributive property.

$f (\frac{5 + \sqrt{5}}{2}) = 3 \cdot 625 + 3 (275 \sqrt{5}) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.13

Multiply $3$ by $625$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 3 (275 \sqrt{5}) - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.14

Multiply $275$ by $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 {(\frac{5 + \sqrt{5}}{2})}^{4} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.15

Apply the product rule to $\frac{5 + \sqrt{5}}{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{{(5 + \sqrt{5})}^{4}}{2^{4}} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.16

Raise $2$ to the power of $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{{(5 + \sqrt{5})}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.17

Use the Binomial Theorem.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{5^{4} + 4 \cdot (5^{3} \sqrt{5}) + 6 \cdot (5^{2} {\sqrt{5}}^{2}) + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18

Simplify each term.

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Step 19.2.1.18.1

Raise $5$ to the power of $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 4 \cdot (5^{3} \sqrt{5}) + 6 \cdot (5^{2} {\sqrt{5}}^{2}) + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.2

Raise $5$ to the power of $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 4 \cdot (125 \sqrt{5}) + 6 \cdot (5^{2} {\sqrt{5}}^{2}) + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.3

Multiply $4$ by $125$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 6 \cdot (5^{2} {\sqrt{5}}^{2}) + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.4

Raise $5$ to the power of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 6 \cdot (25 {\sqrt{5}}^{2}) + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.5

Multiply $6$ by $25$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 150 {\sqrt{5}}^{2} + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 19.2.1.18.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 150 {(5^{\frac{1}{2}})}^{2} + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 150 \cdot 5^{\frac{1}{2} \cdot 2} + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 150 \cdot 5^{\frac{2}{2}} + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.6.4

Cancel the common factor of $2$ .

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Step 19.2.1.18.6.4.1

Cancel the common factor.

Step 19.2.1.18.6.4.2

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 150 \cdot 5 + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.6.5

Evaluate the exponent.

Step 19.2.1.18.7

Multiply $150$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 4 \cdot (5 {\sqrt{5}}^{3}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.8

Multiply $4$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 20 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.9

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 20 \sqrt{5^{3}} + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.10

Raise $5$ to the power of $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 20 \sqrt{125} + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.11

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 19.2.1.18.11.1

Factor $25$ out of $125$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 20 \sqrt{25 (5)} + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.11.2

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 20 \sqrt{5^{2} \cdot 5} + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.12

Pull terms out from under the radical.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 20 (5 \sqrt{5}) + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.13

Multiply $5$ by $20$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 19.2.1.18.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + {(5^{\frac{1}{2}})}^{4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 5^{\frac{1}{2} \cdot 4}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14.3

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 5^{\frac{4}{2}}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14.4

Cancel the common factor of $4$ and $2$ .

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Step 19.2.1.18.14.4.1

Factor $2$ out of $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 5^{\frac{2 \cdot 2}{2}}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14.4.2

Cancel the common factors.

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Step 19.2.1.18.14.4.2.1

Factor $2$ out of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 (1)}}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14.4.2.2

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 \cdot 1}}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14.4.2.3

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 5^{\frac{2}{1}}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.14.4.2.4

Divide $2$ by $1$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 5^{2}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.18.15

Raise $5$ to the power of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{625 + 500 \sqrt{5} + 750 + 100 \sqrt{5} + 25}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.19

Add $625$ and $750$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{1375 + 500 \sqrt{5} + 100 \sqrt{5} + 25}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.20

Add $1375$ and $25$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{1400 + 500 \sqrt{5} + 100 \sqrt{5}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.21

Add $500 \sqrt{5}$ and $100 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{1400 + 600 \sqrt{5}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.22

Cancel the common factor of $1400 + 600 \sqrt{5}$ and $16$ .

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Step 19.2.1.22.1

Factor $8$ out of $1400$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{8 (175) + 600 \sqrt{5}}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.22.2

Factor $8$ out of $600 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{8 (175) + 8 (75 \sqrt{5})}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.22.3

Factor $8$ out of $8 (175) + 8 (75 \sqrt{5})$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{8 (175 + 75 \sqrt{5})}{16} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.22.4

Cancel the common factors.

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Step 19.2.1.22.4.1

Factor $8$ out of $16$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{8 (175 + 75 \sqrt{5})}{8 \cdot 2} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.22.4.2

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{8 (175 + 75 \sqrt{5})}{8 \cdot 2} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.22.4.3

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - 75 \frac{175 + 75 \sqrt{5}}{2} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.23

Combine $- 75$ and $\frac{175 + 75 \sqrt{5}}{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + \frac{- 75 (175 + 75 \sqrt{5})}{2} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.24

Move the negative in front of the fraction.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 300 {(\frac{5 + \sqrt{5}}{2})}^{3} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.25

Apply the product rule to $\frac{5 + \sqrt{5}}{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 300 (\frac{{(5 + \sqrt{5})}^{3}}{2^{3}}) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.26

Raise $2$ to the power of $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 300 (\frac{{(5 + \sqrt{5})}^{3}}{8}) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.27

Cancel the common factor of $4$ .

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Step 19.2.1.27.1

Factor $4$ out of $300$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 4 (75) (\frac{{(5 + \sqrt{5})}^{3}}{8}) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.27.2

Factor $4$ out of $8$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 4 \cdot (75 (\frac{{(5 + \sqrt{5})}^{3}}{4 \cdot 2})) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.27.3

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 4 \cdot (75 (\frac{{(5 + \sqrt{5})}^{3}}{4 \cdot 2})) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.27.4

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 75 (\frac{{(5 + \sqrt{5})}^{3}}{2}) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.28

Combine $75$ and $\frac{{(5 + \sqrt{5})}^{3}}{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 {(5 + \sqrt{5})}^{3}}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.29

Use the Binomial Theorem.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (5^{3} + 3 \cdot (5^{2} \sqrt{5}) + 3 \cdot (5 {\sqrt{5}}^{2}) + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30

Simplify each term.

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Step 19.2.1.30.1

Raise $5$ to the power of $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 3 \cdot (5^{2} \sqrt{5}) + 3 \cdot (5 {\sqrt{5}}^{2}) + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.2

Raise $5$ to the power of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 3 \cdot (25 \sqrt{5}) + 3 \cdot (5 {\sqrt{5}}^{2}) + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.3

Multiply $3$ by $25$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 3 \cdot (5 {\sqrt{5}}^{2}) + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.4

Multiply $3$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 15 {\sqrt{5}}^{2} + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 19.2.1.30.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 15 {(5^{\frac{1}{2}})}^{2} + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 15 \cdot 5^{\frac{1}{2} \cdot 2} + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 15 \cdot 5^{\frac{2}{2}} + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.5.4

Cancel the common factor of $2$ .

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Step 19.2.1.30.5.4.1

Cancel the common factor.

Step 19.2.1.30.5.4.2

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 15 \cdot 5 + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.5.5

Evaluate the exponent.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 15 \cdot 5 + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.6

Multiply $15$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 75 + {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.7

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 75 + \sqrt{5^{3}})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.8

Raise $5$ to the power of $3$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 75 + \sqrt{125})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.9

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 19.2.1.30.9.1

Factor $25$ out of $125$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 75 + \sqrt{25 (5)})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.9.2

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 75 + \sqrt{5^{2} \cdot 5})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.30.10

Pull terms out from under the radical.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (125 + 75 \sqrt{5} + 75 + 5 \sqrt{5})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.31

Add $125$ and $75$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (200 + 75 \sqrt{5} + 5 \sqrt{5})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.32

Add $75 \sqrt{5}$ and $5 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (200 + 80 \sqrt{5})}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.33

Cancel the common factor of $200 + 80 \sqrt{5}$ and $2$ .

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Step 19.2.1.33.1

Factor $2$ out of $75 (200 + 80 \sqrt{5})$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{2 (75 (100 + 40 \sqrt{5}))}{2} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.33.2

Cancel the common factors.

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Step 19.2.1.33.2.1

Factor $2$ out of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{2 (75 (100 + 40 \sqrt{5}))}{2 (1)} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.33.2.2

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{2 (75 (100 + 40 \sqrt{5}))}{2 \cdot 1} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.33.2.3

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + \frac{75 (100 + 40 \sqrt{5})}{1} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.33.2.4

Divide $75 (100 + 40 \sqrt{5})$ by $1$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 75 (100 + 40 \sqrt{5}) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.34

Apply the distributive property.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 75 \cdot 100 + 75 (40 \sqrt{5}) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.35

Multiply $75$ by $100$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 75 (40 \sqrt{5}) - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.36

Multiply $40$ by $75$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 {(\frac{5 + \sqrt{5}}{2})}^{2} - 1$

Step 19.2.1.37

Apply the product rule to $\frac{5 + \sqrt{5}}{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{{(5 + \sqrt{5})}^{2}}{2^{2}} - 1$

Step 19.2.1.38

Raise $2$ to the power of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{{(5 + \sqrt{5})}^{2}}{4} - 1$

Step 19.2.1.39

Rewrite ${(5 + \sqrt{5})}^{2}$ as $(5 + \sqrt{5}) (5 + \sqrt{5})$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{(5 + \sqrt{5}) (5 + \sqrt{5})}{4} - 1$

Step 19.2.1.40

Expand $(5 + \sqrt{5}) (5 + \sqrt{5})$ using the FOIL Method.

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Step 19.2.1.40.1

Apply the distributive property.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{5 (5 + \sqrt{5}) + \sqrt{5} (5 + \sqrt{5})}{4} - 1$

Step 19.2.1.40.2

Apply the distributive property.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{5 \cdot 5 + 5 \sqrt{5} + \sqrt{5} (5 + \sqrt{5})}{4} - 1$

Step 19.2.1.40.3

Apply the distributive property.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{5 \cdot 5 + 5 \sqrt{5} + \sqrt{5} \cdot 5 + \sqrt{5} \sqrt{5}}{4} - 1$

Step 19.2.1.41

Simplify and combine like terms.

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Step 19.2.1.41.1

Simplify each term.

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Step 19.2.1.41.1.1

Multiply $5$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{25 + 5 \sqrt{5} + \sqrt{5} \cdot 5 + \sqrt{5} \sqrt{5}}{4} - 1$

Step 19.2.1.41.1.2

Move $5$ to the left of $\sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{25 + 5 \sqrt{5} + 5 \cdot \sqrt{5} + \sqrt{5} \sqrt{5}}{4} - 1$

Step 19.2.1.41.1.3

Combine using the product rule for radicals.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{25 + 5 \sqrt{5} + 5 \sqrt{5} + \sqrt{5 \cdot 5}}{4} - 1$

Step 19.2.1.41.1.4

Multiply $5$ by $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{25 + 5 \sqrt{5} + 5 \sqrt{5} + \sqrt{25}}{4} - 1$

Step 19.2.1.41.1.5

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{25 + 5 \sqrt{5} + 5 \sqrt{5} + \sqrt{5^{2}}}{4} - 1$

Step 19.2.1.41.1.6

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{25 + 5 \sqrt{5} + 5 \sqrt{5} + 5}{4} - 1$

Step 19.2.1.41.2

Add $25$ and $5$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{30 + 5 \sqrt{5} + 5 \sqrt{5}}{4} - 1$

Step 19.2.1.41.3

Add $5 \sqrt{5}$ and $5 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{30 + 10 \sqrt{5}}{4} - 1$

Step 19.2.1.42

Cancel the common factor of $30 + 10 \sqrt{5}$ and $4$ .

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Step 19.2.1.42.1

Factor $2$ out of $30$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{2 (15) + 10 \sqrt{5}}{4} - 1$

Step 19.2.1.42.2

Factor $2$ out of $10 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{2 (15) + 2 (5 \sqrt{5})}{4} - 1$

Step 19.2.1.42.3

Factor $2$ out of $2 (15) + 2 (5 \sqrt{5})$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{2 (15 + 5 \sqrt{5})}{4} - 1$

Step 19.2.1.42.4

Cancel the common factors.

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Step 19.2.1.42.4.1

Factor $2$ out of $4$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{2 (15 + 5 \sqrt{5})}{2 \cdot 2} - 1$

Step 19.2.1.42.4.2

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{2 (15 + 5 \sqrt{5})}{2 \cdot 2} - 1$

Step 19.2.1.42.4.3

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - 375 \frac{15 + 5 \sqrt{5}}{2} - 1$

Step 19.2.1.43

Combine $- 375$ and $\frac{15 + 5 \sqrt{5}}{2}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} + \frac{- 375 (15 + 5 \sqrt{5})}{2} - 1$

Step 19.2.1.44

Move the negative in front of the fraction.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} - \frac{75 (175 + 75 \sqrt{5})}{2} + 7500 + 3000 \sqrt{5} - \frac{375 (15 + 5 \sqrt{5})}{2} - 1$

Step 19.2.2

Combine the numerators over the common denominator.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 75 (175 + 75 \sqrt{5}) - 375 (15 + 5 \sqrt{5})}{2}$

Step 19.2.3

Simplify each term.

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Step 19.2.3.1

Apply the distributive property.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 75 \cdot 175 - 75 (75 \sqrt{5}) - 375 (15 + 5 \sqrt{5})}{2}$

Step 19.2.3.2

Multiply $- 75$ by $175$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 13125 - 75 (75 \sqrt{5}) - 375 (15 + 5 \sqrt{5})}{2}$

Step 19.2.3.3

Multiply $75$ by $- 75$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 13125 - 5625 \sqrt{5} - 375 (15 + 5 \sqrt{5})}{2}$

Step 19.2.3.4

Apply the distributive property.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 13125 - 5625 \sqrt{5} - 375 \cdot 15 - 375 (5 \sqrt{5})}{2}$

Step 19.2.3.5

Multiply $- 375$ by $15$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 13125 - 5625 \sqrt{5} - 5625 - 375 (5 \sqrt{5})}{2}$

Step 19.2.3.6

Multiply $5$ by $- 375$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 13125 - 5625 \sqrt{5} - 5625 - 1875 \sqrt{5}}{2}$

Step 19.2.4

Simplify terms.

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Step 19.2.4.1

Subtract $5625$ from $- 13125$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 18750 - 5625 \sqrt{5} - 1875 \sqrt{5}}{2}$

Step 19.2.4.2

Subtract $1875 \sqrt{5}$ from $- 5625 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 18750 - 7500 \sqrt{5}}{2}$

Step 19.2.4.3

Cancel the common factor of $- 18750 - 7500 \sqrt{5}$ and $2$ .

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Step 19.2.4.3.1

Factor $2$ out of $- 18750$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{2 \cdot - 9375 - 7500 \sqrt{5}}{2}$

Step 19.2.4.3.2

Factor $2$ out of $- 7500 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{2 \cdot - 9375 + 2 (- 3750 \sqrt{5})}{2}$

Step 19.2.4.3.3

Factor $2$ out of $2 (- 9375) + 2 (- 3750 \sqrt{5})$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{2 (- 9375 - 3750 \sqrt{5})}{2}$

Step 19.2.4.3.4

Cancel the common factors.

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Step 19.2.4.3.4.1

Factor $2$ out of $2$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{2 (- 9375 - 3750 \sqrt{5})}{2 (1)}$

Step 19.2.4.3.4.2

Cancel the common factor.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{2 (- 9375 - 3750 \sqrt{5})}{2 \cdot 1}$

Step 19.2.4.3.4.3

Rewrite the expression.

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 + \frac{- 9375 - 3750 \sqrt{5}}{1}$

Step 19.2.4.3.4.4

Divide $- 9375 - 3750 \sqrt{5}$ by $1$ .

$f (\frac{5 + \sqrt{5}}{2}) = 1875 + 825 \sqrt{5} + 7500 + 3000 \sqrt{5} - 1 - 9375 - 3750 \sqrt{5}$

Step 19.2.4.4

Simplify by adding and subtracting.

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Step 19.2.4.4.1

Add $1875$ and $7500$ .

$f (\frac{5 + \sqrt{5}}{2}) = 9375 + 825 \sqrt{5} + 3000 \sqrt{5} - 1 - 9375 - 3750 \sqrt{5}$

Step 19.2.4.4.2

Subtract $1$ from $9375$ .

$f (\frac{5 + \sqrt{5}}{2}) = 9374 + 825 \sqrt{5} + 3000 \sqrt{5} - 9375 - 3750 \sqrt{5}$

Step 19.2.4.4.3

Subtract $9375$ from $9374$ .

$f (\frac{5 + \sqrt{5}}{2}) = - 1 + 825 \sqrt{5} + 3000 \sqrt{5} - 3750 \sqrt{5}$

Step 19.2.4.5

Add $825 \sqrt{5}$ and $3000 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = - 1 + 3825 \sqrt{5} - 3750 \sqrt{5}$

Step 19.2.4.6

Subtract $3750 \sqrt{5}$ from $3825 \sqrt{5}$ .

$f (\frac{5 + \sqrt{5}}{2}) = - 1 + 75 \sqrt{5}$

Step 19.2.5

The final answer is $- 1 + 75 \sqrt{5}$ .

$y = - 1 + 75 \sqrt{5}$

Step 20

Evaluate the second derivative at $x = \frac{5 - \sqrt{5}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$120 {(\frac{5 - \sqrt{5}}{2})}^{3} - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21

Evaluate the second derivative.

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Step 21.1

Simplify each term.

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Step 21.1.1

Apply the product rule to $\frac{5 - \sqrt{5}}{2}$ .

$120 \frac{{(5 - \sqrt{5})}^{3}}{2^{3}} - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.2

Raise $2$ to the power of $3$ .

$120 \frac{{(5 - \sqrt{5})}^{3}}{8} - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.3

Cancel the common factor of $8$ .

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Step 21.1.3.1

Factor $8$ out of $120$ .

$8 (15) \frac{{(5 - \sqrt{5})}^{3}}{8} - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.3.2

Cancel the common factor.

$8 \cdot 15 \frac{{(5 - \sqrt{5})}^{3}}{8} - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.3.3

Rewrite the expression.

$15 {(5 - \sqrt{5})}^{3} - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.4

Use the Binomial Theorem.

$15 (5^{3} + 3 \cdot 5^{2} (- \sqrt{5}) + 3 \cdot 5 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5

Simplify each term.

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Step 21.1.5.1

Raise $5$ to the power of $3$ .

$15 (125 + 3 \cdot 5^{2} (- \sqrt{5}) + 3 \cdot 5 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.2

Raise $5$ to the power of $2$ .

$15 (125 + 3 \cdot 25 (- \sqrt{5}) + 3 \cdot 5 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.3

Multiply $3$ by $25$ .

$15 (125 + 75 (- \sqrt{5}) + 3 \cdot 5 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.4

Multiply $- 1$ by $75$ .

$15 (125 - 75 \sqrt{5} + 3 \cdot 5 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.5

Multiply $3$ by $5$ .

$15 (125 - 75 \sqrt{5} + 15 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.6

Apply the product rule to $- \sqrt{5}$ .

$15 (125 - 75 \sqrt{5} + 15 ({(- 1)}^{2} {\sqrt{5}}^{2}) + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.7

Raise $- 1$ to the power of $2$ .

$15 (125 - 75 \sqrt{5} + 15 (1 {\sqrt{5}}^{2}) + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.8

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$15 (125 - 75 \sqrt{5} + 15 {\sqrt{5}}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.9

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 21.1.5.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$15 (125 - 75 \sqrt{5} + 15 {(5^{\frac{1}{2}})}^{2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$15 (125 - 75 \sqrt{5} + 15 \cdot 5^{\frac{1}{2} \cdot 2} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.9.3

Combine $\frac{1}{2}$ and $2$ .

$15 (125 - 75 \sqrt{5} + 15 \cdot 5^{\frac{2}{2}} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.9.4

Cancel the common factor of $2$ .

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Step 21.1.5.9.4.1

Cancel the common factor.

$15 (125 - 75 \sqrt{5} + 15 \cdot 5^{\frac{2}{2}} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.9.4.2

Rewrite the expression.

$15 (125 - 75 \sqrt{5} + 15 \cdot 5^{1} + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.9.5

Evaluate the exponent.

$15 (125 - 75 \sqrt{5} + 15 \cdot 5 + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.10

Multiply $15$ by $5$ .

$15 (125 - 75 \sqrt{5} + 75 + {(- \sqrt{5})}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.11

Apply the product rule to $- \sqrt{5}$ .

$15 (125 - 75 \sqrt{5} + 75 + {(- 1)}^{3} {\sqrt{5}}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.12

Raise $- 1$ to the power of $3$ .

$15 (125 - 75 \sqrt{5} + 75 - {\sqrt{5}}^{3}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.13

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$15 (125 - 75 \sqrt{5} + 75 - \sqrt{5^{3}}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.14

Raise $5$ to the power of $3$ .

$15 (125 - 75 \sqrt{5} + 75 - \sqrt{125}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.15

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 21.1.5.15.1

Factor $25$ out of $125$ .

$15 (125 - 75 \sqrt{5} + 75 - \sqrt{25 (5)}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.15.2

Rewrite $25$ as $5^{2}$ .

$15 (125 - 75 \sqrt{5} + 75 - \sqrt{5^{2} \cdot 5}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.16

Pull terms out from under the radical.

$15 (125 - 75 \sqrt{5} + 75 - (5 \sqrt{5})) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.5.17

Multiply $5$ by $- 1$ .

$15 (125 - 75 \sqrt{5} + 75 - 5 \sqrt{5}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.6

Add $125$ and $75$ .

$15 (200 - 75 \sqrt{5} - 5 \sqrt{5}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.7

Subtract $5 \sqrt{5}$ from $- 75 \sqrt{5}$ .

$15 (200 - 80 \sqrt{5}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.8

Apply the distributive property.

$15 \cdot 200 + 15 (- 80 \sqrt{5}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.9

Multiply $15$ by $200$ .

$3000 + 15 (- 80 \sqrt{5}) - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.10

Multiply $- 80$ by $15$ .

$3000 - 1200 \sqrt{5} - 900 {(\frac{5 - \sqrt{5}}{2})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.11

Apply the product rule to $\frac{5 - \sqrt{5}}{2}$ .

$3000 - 1200 \sqrt{5} - 900 \frac{{(5 - \sqrt{5})}^{2}}{2^{2}} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.12

Raise $2$ to the power of $2$ .

$3000 - 1200 \sqrt{5} - 900 \frac{{(5 - \sqrt{5})}^{2}}{4} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.13

Cancel the common factor of $4$ .

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Step 21.1.13.1

Factor $4$ out of $- 900$ .

$3000 - 1200 \sqrt{5} + 4 (- 225) \frac{{(5 - \sqrt{5})}^{2}}{4} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.13.2

Cancel the common factor.

$3000 - 1200 \sqrt{5} + 4 \cdot - 225 \frac{{(5 - \sqrt{5})}^{2}}{4} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.13.3

Rewrite the expression.

$3000 - 1200 \sqrt{5} - 225 {(5 - \sqrt{5})}^{2} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.14

Rewrite ${(5 - \sqrt{5})}^{2}$ as $(5 - \sqrt{5}) (5 - \sqrt{5})$ .

$3000 - 1200 \sqrt{5} - 225 ((5 - \sqrt{5}) (5 - \sqrt{5})) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.15

Expand $(5 - \sqrt{5}) (5 - \sqrt{5})$ using the FOIL Method.

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Step 21.1.15.1

Apply the distributive property.

$3000 - 1200 \sqrt{5} - 225 (5 (5 - \sqrt{5}) - \sqrt{5} (5 - \sqrt{5})) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.15.2

Apply the distributive property.

$3000 - 1200 \sqrt{5} - 225 (5 \cdot 5 + 5 (- \sqrt{5}) - \sqrt{5} (5 - \sqrt{5})) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.15.3

Apply the distributive property.

$3000 - 1200 \sqrt{5} - 225 (5 \cdot 5 + 5 (- \sqrt{5}) - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{5})) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16

Simplify and combine like terms.

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Step 21.1.16.1

Simplify each term.

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Step 21.1.16.1.1

Multiply $5$ by $5$ .

$3000 - 1200 \sqrt{5} - 225 (25 + 5 (- \sqrt{5}) - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{5})) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.2

Multiply $- 1$ by $5$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{5})) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.3

Multiply $5$ by $- 1$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} - \sqrt{5} (- \sqrt{5})) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.4

Multiply $- \sqrt{5} (- \sqrt{5})$ .

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Step 21.1.16.1.4.1

Multiply $- 1$ by $- 1$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + 1 \sqrt{5} \sqrt{5}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.4.2

Multiply $\sqrt{5}$ by $1$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + \sqrt{5} \sqrt{5}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.4.4

Raise $\sqrt{5}$ to the power of $1$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + {\sqrt{5}}^{1 + 1}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.4.6

Add $1$ and $1$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + {\sqrt{5}}^{2}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 21.1.16.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + {(5^{\frac{1}{2}})}^{2}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + 5^{\frac{1}{2} \cdot 2}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + 5^{\frac{2}{2}}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.5.4

Cancel the common factor of $2$ .

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Step 21.1.16.1.5.4.1

Cancel the common factor.

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + 5^{\frac{2}{2}}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.5.4.2

Rewrite the expression.

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + 5^{1}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.1.5.5

Evaluate the exponent.

$3000 - 1200 \sqrt{5} - 225 (25 - 5 \sqrt{5} - 5 \sqrt{5} + 5) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.2

Add $25$ and $5$ .

$3000 - 1200 \sqrt{5} - 225 (30 - 5 \sqrt{5} - 5 \sqrt{5}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.16.3

Subtract $5 \sqrt{5}$ from $- 5 \sqrt{5}$ .

$3000 - 1200 \sqrt{5} - 225 (30 - 10 \sqrt{5}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.17

Apply the distributive property.

$3000 - 1200 \sqrt{5} - 225 \cdot 30 - 225 (- 10 \sqrt{5}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.18

Multiply $- 225$ by $30$ .

$3000 - 1200 \sqrt{5} - 6750 - 225 (- 10 \sqrt{5}) + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.19

Multiply $- 10$ by $- 225$ .

$3000 - 1200 \sqrt{5} - 6750 + 2250 \sqrt{5} + 1800 (\frac{5 - \sqrt{5}}{2}) - 750$

Step 21.1.20

Cancel the common factor of $2$ .

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Step 21.1.20.1

Factor $2$ out of $1800$ .

$3000 - 1200 \sqrt{5} - 6750 + 2250 \sqrt{5} + 2 (900) \frac{5 - \sqrt{5}}{2} - 750$

Step 21.1.20.2

Cancel the common factor.

$3000 - 1200 \sqrt{5} - 6750 + 2250 \sqrt{5} + 2 \cdot 900 \frac{5 - \sqrt{5}}{2} - 750$

Step 21.1.20.3

Rewrite the expression.

$3000 - 1200 \sqrt{5} - 6750 + 2250 \sqrt{5} + 900 (5 - \sqrt{5}) - 750$

Step 21.1.21

Apply the distributive property.

$3000 - 1200 \sqrt{5} - 6750 + 2250 \sqrt{5} + 900 \cdot 5 + 900 (- \sqrt{5}) - 750$

Step 21.1.22

Multiply $900$ by $5$ .

$3000 - 1200 \sqrt{5} - 6750 + 2250 \sqrt{5} + 4500 + 900 (- \sqrt{5}) - 750$

Step 21.1.23

Multiply $- 1$ by $900$ .

$3000 - 1200 \sqrt{5} - 6750 + 2250 \sqrt{5} + 4500 - 900 \sqrt{5} - 750$

Step 21.2

Simplify by adding terms.

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Step 21.2.1

Subtract $6750$ from $3000$ .

$- 3750 - 1200 \sqrt{5} + 2250 \sqrt{5} + 4500 - 900 \sqrt{5} - 750$

Step 21.2.2

Simplify by adding and subtracting.

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Step 21.2.2.1

Add $- 3750$ and $4500$ .

$750 - 1200 \sqrt{5} + 2250 \sqrt{5} - 900 \sqrt{5} - 750$

Step 21.2.2.2

Subtract $750$ from $750$ .

$0 - 1200 \sqrt{5} + 2250 \sqrt{5} - 900 \sqrt{5}$

Step 21.2.2.3

Subtract $1200 \sqrt{5}$ from $0$ .

$- 1200 \sqrt{5} + 2250 \sqrt{5} - 900 \sqrt{5}$

Step 21.2.3

Add $- 1200 \sqrt{5}$ and $2250 \sqrt{5}$ .

$1050 \sqrt{5} - 900 \sqrt{5}$

Step 21.2.4

Subtract $900 \sqrt{5}$ from $1050 \sqrt{5}$ .

$150 \sqrt{5}$

Step 22

$x = \frac{5 - \sqrt{5}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{5 - \sqrt{5}}{2}$ is a local minimum

Step 23

Find the y-value when $x = \frac{5 - \sqrt{5}}{2}$ .

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Step 23.1

Replace the variable $x$ with $\frac{5 - \sqrt{5}}{2}$ in the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 6 {(\frac{5 - \sqrt{5}}{2})}^{5} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2

Simplify the result.

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Step 23.2.1

Simplify each term.

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Step 23.2.1.1

Apply the product rule to $\frac{5 - \sqrt{5}}{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 6 (\frac{{(5 - \sqrt{5})}^{5}}{2^{5}}) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.2

Raise $2$ to the power of $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 6 (\frac{{(5 - \sqrt{5})}^{5}}{32}) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.3

Cancel the common factor of $2$ .

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Step 23.2.1.3.1

Factor $2$ out of $6$ .

$f (\frac{5 - \sqrt{5}}{2}) = 2 (3) (\frac{{(5 - \sqrt{5})}^{5}}{32}) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.3.2

Factor $2$ out of $32$ .

$f (\frac{5 - \sqrt{5}}{2}) = 2 \cdot (3 (\frac{{(5 - \sqrt{5})}^{5}}{2 \cdot 16})) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.3.3

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 2 \cdot (3 (\frac{{(5 - \sqrt{5})}^{5}}{2 \cdot 16})) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.3.4

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 3 (\frac{{(5 - \sqrt{5})}^{5}}{16}) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.4

Combine $3$ and $\frac{{(5 - \sqrt{5})}^{5}}{16}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 {(5 - \sqrt{5})}^{5}}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.5

Use the Binomial Theorem.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (5^{5} + 5 \cdot (5^{4} (- \sqrt{5})) + 10 \cdot (5^{3} {(- \sqrt{5})}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6

Simplify each term.

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Step 23.2.1.6.1

Raise $5$ to the power of $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 + 5 \cdot (5^{4} (- \sqrt{5})) + 10 \cdot (5^{3} {(- \sqrt{5})}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.2

Multiply $5$ by $5^{4}$ by adding the exponents.

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Step 23.2.1.6.2.1

Multiply $5$ by $5^{4}$ .

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Step 23.2.1.6.2.1.1

Raise $5$ to the power of $1$ .

Step 23.2.1.6.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 + 5^{1 + 4} (- \sqrt{5}) + 10 \cdot (5^{3} {(- \sqrt{5})}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.2.2

Add $1$ and $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 + 5^{5} (- \sqrt{5}) + 10 \cdot (5^{3} {(- \sqrt{5})}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.3

Raise $5$ to the power of $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 + 3125 (- \sqrt{5}) + 10 \cdot (5^{3} {(- \sqrt{5})}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.4

Multiply $- 1$ by $3125$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 10 \cdot (5^{3} {(- \sqrt{5})}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.5

Raise $5$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 10 \cdot (125 {(- \sqrt{5})}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.6

Multiply $10$ by $125$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 {(- \sqrt{5})}^{2} + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.7

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 ({(- 1)}^{2} {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.8

Raise $- 1$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 (1 {\sqrt{5}}^{2}) + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.9

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 {\sqrt{5}}^{2} + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.10

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 23.2.1.6.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 {(5^{\frac{1}{2}})}^{2} + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 \cdot 5^{\frac{1}{2} \cdot 2} + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.10.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 \cdot 5^{\frac{2}{2}} + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.10.4

Cancel the common factor of $2$ .

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Step 23.2.1.6.10.4.1

Cancel the common factor.

Step 23.2.1.6.10.4.2

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 1250 \cdot 5 + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.10.5

Evaluate the exponent.

Step 23.2.1.6.11

Multiply $1250$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 10 \cdot (5^{2} {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.12

Raise $5$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 10 \cdot (25 {(- \sqrt{5})}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.13

Multiply $10$ by $25$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 {(- \sqrt{5})}^{3} + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.14

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 ({(- 1)}^{3} {\sqrt{5}}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.15

Raise $- 1$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 (- {\sqrt{5}}^{3}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.16

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 (- \sqrt{5^{3}}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.17

Raise $5$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 (- \sqrt{125}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.18

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 23.2.1.6.18.1

Factor $25$ out of $125$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 (- \sqrt{25 (5)}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.18.2

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 (- \sqrt{5^{2} \cdot 5}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.19

Pull terms out from under the radical.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 (- (5 \sqrt{5})) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.20

Multiply $5$ by $- 1$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 + 250 (- 5 \sqrt{5}) + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.21

Multiply $- 5$ by $250$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 5 \cdot (5 {(- \sqrt{5})}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.22

Multiply $5$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 {(- \sqrt{5})}^{4} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.23

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 ({(- 1)}^{4} {\sqrt{5}}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.24

Raise $- 1$ to the power of $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 (1 {\sqrt{5}}^{4}) + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.25

Multiply ${\sqrt{5}}^{4}$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 {\sqrt{5}}^{4} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 23.2.1.6.26.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 {(5^{\frac{1}{2}})}^{4} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 5^{\frac{1}{2} \cdot 4} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26.3

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 5^{\frac{4}{2}} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26.4

Cancel the common factor of $4$ and $2$ .

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Step 23.2.1.6.26.4.1

Factor $2$ out of $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 5^{\frac{2 \cdot 2}{2}} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26.4.2

Cancel the common factors.

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Step 23.2.1.6.26.4.2.1

Factor $2$ out of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 5^{\frac{2 \cdot 2}{2 (1)}} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26.4.2.2

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 5^{\frac{2 \cdot 2}{2 \cdot 1}} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26.4.2.3

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 5^{\frac{2}{1}} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.26.4.2.4

Divide $2$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 5^{2} + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.27

Raise $5$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 25 \cdot 25 + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.28

Multiply $25$ by $25$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 + {(- \sqrt{5})}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.29

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 + {(- 1)}^{5} {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.30

Raise $- 1$ to the power of $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 - {\sqrt{5}}^{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.31

Rewrite ${\sqrt{5}}^{5}$ as $\sqrt{5^{5}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 - \sqrt{5^{5}})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.32

Raise $5$ to the power of $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 - \sqrt{3125})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.33

Rewrite $3125$ as $25^{2} \cdot 5$ .

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Step 23.2.1.6.33.1

Factor $625$ out of $3125$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 - \sqrt{625 (5)})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.33.2

Rewrite $625$ as $25^{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 - \sqrt{25^{2} \cdot 5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.34

Pull terms out from under the radical.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 - (25 \sqrt{5}))}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.6.35

Multiply $25$ by $- 1$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (3125 - 3125 \sqrt{5} + 6250 - 1250 \sqrt{5} + 625 - 25 \sqrt{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.7

Add $3125$ and $6250$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (9375 - 3125 \sqrt{5} - 1250 \sqrt{5} + 625 - 25 \sqrt{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.8

Add $9375$ and $625$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (10000 - 3125 \sqrt{5} - 1250 \sqrt{5} - 25 \sqrt{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.9

Subtract $1250 \sqrt{5}$ from $- 3125 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (10000 - 4375 \sqrt{5} - 25 \sqrt{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.10

Subtract $25 \sqrt{5}$ from $- 4375 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (10000 - 4400 \sqrt{5})}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.11

Cancel the common factor of $10000 - 4400 \sqrt{5}$ and $16$ .

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Step 23.2.1.11.1

Factor $16$ out of $3 (10000 - 4400 \sqrt{5})$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{16 (3 (625 - 275 \sqrt{5}))}{16} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.11.2

Cancel the common factors.

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Step 23.2.1.11.2.1

Factor $16$ out of $16$ .

$f (\frac{5 - \sqrt{5}}{2}) = \frac{16 (3 (625 - 275 \sqrt{5}))}{16 (1)} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.11.2.2

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{16 (3 (625 - 275 \sqrt{5}))}{16 \cdot 1} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.11.2.3

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = \frac{3 (625 - 275 \sqrt{5})}{1} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.11.2.4

Divide $3 (625 - 275 \sqrt{5})$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 3 (625 - 275 \sqrt{5}) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.12

Apply the distributive property.

$f (\frac{5 - \sqrt{5}}{2}) = 3 \cdot 625 + 3 (- 275 \sqrt{5}) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.13

Multiply $3$ by $625$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 + 3 (- 275 \sqrt{5}) - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.14

Multiply $- 275$ by $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 {(\frac{5 - \sqrt{5}}{2})}^{4} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.15

Apply the product rule to $\frac{5 - \sqrt{5}}{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{{(5 - \sqrt{5})}^{4}}{2^{4}} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.16

Raise $2$ to the power of $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{{(5 - \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.17

Use the Binomial Theorem.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{5^{4} + 4 \cdot (5^{3} (- \sqrt{5})) + 6 \cdot (5^{2} {(- \sqrt{5})}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18

Simplify each term.

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Step 23.2.1.18.1

Raise $5$ to the power of $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 + 4 \cdot (5^{3} (- \sqrt{5})) + 6 \cdot (5^{2} {(- \sqrt{5})}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.2

Raise $5$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 + 4 \cdot (125 (- \sqrt{5})) + 6 \cdot (5^{2} {(- \sqrt{5})}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.3

Multiply $4$ by $125$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 + 500 (- \sqrt{5}) + 6 \cdot (5^{2} {(- \sqrt{5})}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.4

Multiply $- 1$ by $500$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 6 \cdot (5^{2} {(- \sqrt{5})}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.5

Raise $5$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 6 \cdot (25 {(- \sqrt{5})}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.6

Multiply $6$ by $25$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 {(- \sqrt{5})}^{2} + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.7

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 ({(- 1)}^{2} {\sqrt{5}}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.8

Raise $- 1$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 (1 {\sqrt{5}}^{2}) + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.9

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 {\sqrt{5}}^{2} + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.10

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 23.2.1.18.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 {(5^{\frac{1}{2}})}^{2} + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 \cdot 5^{\frac{1}{2} \cdot 2} + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.10.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 \cdot 5^{\frac{2}{2}} + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.10.4

Cancel the common factor of $2$ .

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Step 23.2.1.18.10.4.1

Cancel the common factor.

Step 23.2.1.18.10.4.2

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 150 \cdot 5 + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.10.5

Evaluate the exponent.

Step 23.2.1.18.11

Multiply $150$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 4 \cdot (5 {(- \sqrt{5})}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.12

Multiply $4$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.13

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 ({(- 1)}^{3} {\sqrt{5}}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.14

Raise $- 1$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 (- {\sqrt{5}}^{3}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.15

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 (- \sqrt{5^{3}}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.16

Raise $5$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 (- \sqrt{125}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.17

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 23.2.1.18.17.1

Factor $25$ out of $125$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 (- \sqrt{25 (5)}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.17.2

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 (- \sqrt{5^{2} \cdot 5}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.18

Pull terms out from under the radical.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 (- (5 \sqrt{5})) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.19

Multiply $5$ by $- 1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 + 20 (- 5 \sqrt{5}) + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.20

Multiply $- 5$ by $20$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + {(- \sqrt{5})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.21

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + {(- 1)}^{4} {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.22

Raise $- 1$ to the power of $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 1 {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.23

Multiply ${\sqrt{5}}^{4}$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + {\sqrt{5}}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 23.2.1.18.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + {(5^{\frac{1}{2}})}^{4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 5^{\frac{1}{2} \cdot 4}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24.3

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 5^{\frac{4}{2}}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24.4

Cancel the common factor of $4$ and $2$ .

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Step 23.2.1.18.24.4.1

Factor $2$ out of $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 5^{\frac{2 \cdot 2}{2}}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24.4.2

Cancel the common factors.

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Step 23.2.1.18.24.4.2.1

Factor $2$ out of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 (1)}}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24.4.2.2

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 \cdot 1}}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24.4.2.3

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 5^{\frac{2}{1}}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.24.4.2.4

Divide $2$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 5^{2}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.18.25

Raise $5$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{625 - 500 \sqrt{5} + 750 - 100 \sqrt{5} + 25}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.19

Add $625$ and $750$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{1375 - 500 \sqrt{5} - 100 \sqrt{5} + 25}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.20

Add $1375$ and $25$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{1400 - 500 \sqrt{5} - 100 \sqrt{5}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.21

Subtract $100 \sqrt{5}$ from $- 500 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{1400 - 600 \sqrt{5}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.22

Cancel the common factor of $1400 - 600 \sqrt{5}$ and $16$ .

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Step 23.2.1.22.1

Factor $8$ out of $1400$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{8 (175) - 600 \sqrt{5}}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.22.2

Factor $8$ out of $- 600 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{8 (175) + 8 (- 75 \sqrt{5})}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.22.3

Factor $8$ out of $8 (175) + 8 (- 75 \sqrt{5})$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{8 (175 - 75 \sqrt{5})}{16} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.22.4

Cancel the common factors.

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Step 23.2.1.22.4.1

Factor $8$ out of $16$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{8 (175 - 75 \sqrt{5})}{8 \cdot 2} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.22.4.2

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{8 (175 - 75 \sqrt{5})}{8 \cdot 2} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.22.4.3

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - 75 \frac{175 - 75 \sqrt{5}}{2} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.23

Combine $- 75$ and $\frac{175 - 75 \sqrt{5}}{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + \frac{- 75 (175 - 75 \sqrt{5})}{2} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.24

Move the negative in front of the fraction.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 300 {(\frac{5 - \sqrt{5}}{2})}^{3} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.25

Apply the product rule to $\frac{5 - \sqrt{5}}{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 300 (\frac{{(5 - \sqrt{5})}^{3}}{2^{3}}) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.26

Raise $2$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 300 (\frac{{(5 - \sqrt{5})}^{3}}{8}) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.27

Cancel the common factor of $4$ .

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Step 23.2.1.27.1

Factor $4$ out of $300$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 4 (75) (\frac{{(5 - \sqrt{5})}^{3}}{8}) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.27.2

Factor $4$ out of $8$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 4 \cdot (75 (\frac{{(5 - \sqrt{5})}^{3}}{4 \cdot 2})) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.27.3

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 4 \cdot (75 (\frac{{(5 - \sqrt{5})}^{3}}{4 \cdot 2})) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.27.4

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 75 (\frac{{(5 - \sqrt{5})}^{3}}{2}) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.28

Combine $75$ and $\frac{{(5 - \sqrt{5})}^{3}}{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 {(5 - \sqrt{5})}^{3}}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.29

Use the Binomial Theorem.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (5^{3} + 3 \cdot (5^{2} (- \sqrt{5})) + 3 \cdot (5 {(- \sqrt{5})}^{2}) + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30

Simplify each term.

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Step 23.2.1.30.1

Raise $5$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 + 3 \cdot (5^{2} (- \sqrt{5})) + 3 \cdot (5 {(- \sqrt{5})}^{2}) + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.2

Raise $5$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 + 3 \cdot (25 (- \sqrt{5})) + 3 \cdot (5 {(- \sqrt{5})}^{2}) + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.3

Multiply $3$ by $25$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 + 75 (- \sqrt{5}) + 3 \cdot (5 {(- \sqrt{5})}^{2}) + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.4

Multiply $- 1$ by $75$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 3 \cdot (5 {(- \sqrt{5})}^{2}) + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.5

Multiply $3$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.6

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 ({(- 1)}^{2} {\sqrt{5}}^{2}) + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.7

Raise $- 1$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 (1 {\sqrt{5}}^{2}) + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.8

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 {\sqrt{5}}^{2} + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.9

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 23.2.1.30.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 {(5^{\frac{1}{2}})}^{2} + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 \cdot 5^{\frac{1}{2} \cdot 2} + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.9.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 \cdot 5^{\frac{2}{2}} + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.9.4

Cancel the common factor of $2$ .

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Step 23.2.1.30.9.4.1

Cancel the common factor.

Step 23.2.1.30.9.4.2

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 \cdot 5 + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.9.5

Evaluate the exponent.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 15 \cdot 5 + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.10

Multiply $15$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 + {(- \sqrt{5})}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.11

Apply the product rule to $- \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 + {(- 1)}^{3} {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.12

Raise $- 1$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 - {\sqrt{5}}^{3})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.13

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 - \sqrt{5^{3}})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.14

Raise $5$ to the power of $3$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 - \sqrt{125})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.15

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 23.2.1.30.15.1

Factor $25$ out of $125$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 - \sqrt{25 (5)})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.15.2

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 - \sqrt{5^{2} \cdot 5})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.16

Pull terms out from under the radical.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 - (5 \sqrt{5}))}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.30.17

Multiply $5$ by $- 1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (125 - 75 \sqrt{5} + 75 - 5 \sqrt{5})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.31

Add $125$ and $75$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (200 - 75 \sqrt{5} - 5 \sqrt{5})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.32

Subtract $5 \sqrt{5}$ from $- 75 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (200 - 80 \sqrt{5})}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.33

Cancel the common factor of $200 - 80 \sqrt{5}$ and $2$ .

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Step 23.2.1.33.1

Factor $2$ out of $75 (200 - 80 \sqrt{5})$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{2 (75 (100 - 40 \sqrt{5}))}{2} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.33.2

Cancel the common factors.

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Step 23.2.1.33.2.1

Factor $2$ out of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{2 (75 (100 - 40 \sqrt{5}))}{2 (1)} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.33.2.2

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{2 (75 (100 - 40 \sqrt{5}))}{2 \cdot 1} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.33.2.3

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + \frac{75 (100 - 40 \sqrt{5})}{1} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.33.2.4

Divide $75 (100 - 40 \sqrt{5})$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 75 (100 - 40 \sqrt{5}) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.34

Apply the distributive property.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 75 \cdot 100 + 75 (- 40 \sqrt{5}) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.35

Multiply $75$ by $100$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 + 75 (- 40 \sqrt{5}) - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.36

Multiply $- 40$ by $75$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 {(\frac{5 - \sqrt{5}}{2})}^{2} - 1$

Step 23.2.1.37

Apply the product rule to $\frac{5 - \sqrt{5}}{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{{(5 - \sqrt{5})}^{2}}{2^{2}} - 1$

Step 23.2.1.38

Raise $2$ to the power of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{{(5 - \sqrt{5})}^{2}}{4} - 1$

Step 23.2.1.39

Rewrite ${(5 - \sqrt{5})}^{2}$ as $(5 - \sqrt{5}) (5 - \sqrt{5})$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{(5 - \sqrt{5}) (5 - \sqrt{5})}{4} - 1$

Step 23.2.1.40

Expand $(5 - \sqrt{5}) (5 - \sqrt{5})$ using the FOIL Method.

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Step 23.2.1.40.1

Apply the distributive property.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{5 (5 - \sqrt{5}) - \sqrt{5} (5 - \sqrt{5})}{4} - 1$

Step 23.2.1.40.2

Apply the distributive property.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{5 \cdot 5 + 5 (- \sqrt{5}) - \sqrt{5} (5 - \sqrt{5})}{4} - 1$

Step 23.2.1.40.3

Apply the distributive property.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{5 \cdot 5 + 5 (- \sqrt{5}) - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{5})}{4} - 1$

Step 23.2.1.41

Simplify and combine like terms.

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Step 23.2.1.41.1

Simplify each term.

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Step 23.2.1.41.1.1

Multiply $5$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 + 5 (- \sqrt{5}) - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{5})}{4} - 1$

Step 23.2.1.41.1.2

Multiply $- 1$ by $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{5})}{4} - 1$

Step 23.2.1.41.1.3

Multiply $5$ by $- 1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} - \sqrt{5} (- \sqrt{5})}{4} - 1$

Step 23.2.1.41.1.4

Multiply $- \sqrt{5} (- \sqrt{5})$ .

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Step 23.2.1.41.1.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + 1 \sqrt{5} \sqrt{5}}{4} - 1$

Step 23.2.1.41.1.4.2

Multiply $\sqrt{5}$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + \sqrt{5} \sqrt{5}}{4} - 1$

Step 23.2.1.41.1.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + \sqrt{5} \sqrt{5}}{4} - 1$

Step 23.2.1.41.1.4.4

Raise $\sqrt{5}$ to the power of $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + \sqrt{5} \sqrt{5}}{4} - 1$

Step 23.2.1.41.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + {\sqrt{5}}^{1 + 1}}{4} - 1$

Step 23.2.1.41.1.4.6

Add $1$ and $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + {\sqrt{5}}^{2}}{4} - 1$

Step 23.2.1.41.1.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 23.2.1.41.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + {(5^{\frac{1}{2}})}^{2}}{4} - 1$

Step 23.2.1.41.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + 5^{\frac{1}{2} \cdot 2}}{4} - 1$

Step 23.2.1.41.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + 5^{\frac{2}{2}}}{4} - 1$

Step 23.2.1.41.1.5.4

Cancel the common factor of $2$ .

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Step 23.2.1.41.1.5.4.1

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + 5^{\frac{2}{2}}}{4} - 1$

Step 23.2.1.41.1.5.4.2

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + 5}{4} - 1$

Step 23.2.1.41.1.5.5

Evaluate the exponent.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{25 - 5 \sqrt{5} - 5 \sqrt{5} + 5}{4} - 1$

Step 23.2.1.41.2

Add $25$ and $5$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{30 - 5 \sqrt{5} - 5 \sqrt{5}}{4} - 1$

Step 23.2.1.41.3

Subtract $5 \sqrt{5}$ from $- 5 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{30 - 10 \sqrt{5}}{4} - 1$

Step 23.2.1.42

Cancel the common factor of $30 - 10 \sqrt{5}$ and $4$ .

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Step 23.2.1.42.1

Factor $2$ out of $30$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{2 (15) - 10 \sqrt{5}}{4} - 1$

Step 23.2.1.42.2

Factor $2$ out of $- 10 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{2 (15) + 2 (- 5 \sqrt{5})}{4} - 1$

Step 23.2.1.42.3

Factor $2$ out of $2 (15) + 2 (- 5 \sqrt{5})$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{2 (15 - 5 \sqrt{5})}{4} - 1$

Step 23.2.1.42.4

Cancel the common factors.

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Step 23.2.1.42.4.1

Factor $2$ out of $4$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{2 (15 - 5 \sqrt{5})}{2 \cdot 2} - 1$

Step 23.2.1.42.4.2

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{2 (15 - 5 \sqrt{5})}{2 \cdot 2} - 1$

Step 23.2.1.42.4.3

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - 375 \frac{15 - 5 \sqrt{5}}{2} - 1$

Step 23.2.1.43

Combine $- 375$ and $\frac{15 - 5 \sqrt{5}}{2}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} + \frac{- 375 (15 - 5 \sqrt{5})}{2} - 1$

Step 23.2.1.44

Move the negative in front of the fraction.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} - \frac{75 (175 - 75 \sqrt{5})}{2} + 7500 - 3000 \sqrt{5} - \frac{375 (15 - 5 \sqrt{5})}{2} - 1$

Step 23.2.2

Combine the numerators over the common denominator.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 75 (175 - 75 \sqrt{5}) - 375 (15 - 5 \sqrt{5})}{2}$

Step 23.2.3

Simplify each term.

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Step 23.2.3.1

Apply the distributive property.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 75 \cdot 175 - 75 (- 75 \sqrt{5}) - 375 (15 - 5 \sqrt{5})}{2}$

Step 23.2.3.2

Multiply $- 75$ by $175$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 13125 - 75 (- 75 \sqrt{5}) - 375 (15 - 5 \sqrt{5})}{2}$

Step 23.2.3.3

Multiply $- 75$ by $- 75$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 13125 + 5625 \sqrt{5} - 375 (15 - 5 \sqrt{5})}{2}$

Step 23.2.3.4

Apply the distributive property.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 13125 + 5625 \sqrt{5} - 375 \cdot 15 - 375 (- 5 \sqrt{5})}{2}$

Step 23.2.3.5

Multiply $- 375$ by $15$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 13125 + 5625 \sqrt{5} - 5625 - 375 (- 5 \sqrt{5})}{2}$

Step 23.2.3.6

Multiply $- 5$ by $- 375$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 13125 + 5625 \sqrt{5} - 5625 + 1875 \sqrt{5}}{2}$

Step 23.2.4

Simplify terms.

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Step 23.2.4.1

Subtract $5625$ from $- 13125$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 18750 + 5625 \sqrt{5} + 1875 \sqrt{5}}{2}$

Step 23.2.4.2

Add $5625 \sqrt{5}$ and $1875 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 18750 + 7500 \sqrt{5}}{2}$

Step 23.2.4.3

Cancel the common factor of $- 18750 + 7500 \sqrt{5}$ and $2$ .

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Step 23.2.4.3.1

Factor $2$ out of $- 18750$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{2 \cdot - 9375 + 7500 \sqrt{5}}{2}$

Step 23.2.4.3.2

Factor $2$ out of $7500 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{2 \cdot - 9375 + 2 (3750 \sqrt{5})}{2}$

Step 23.2.4.3.3

Factor $2$ out of $2 (- 9375) + 2 (3750 \sqrt{5})$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{2 (- 9375 + 3750 \sqrt{5})}{2}$

Step 23.2.4.3.4

Cancel the common factors.

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Step 23.2.4.3.4.1

Factor $2$ out of $2$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{2 (- 9375 + 3750 \sqrt{5})}{2 (1)}$

Step 23.2.4.3.4.2

Cancel the common factor.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{2 (- 9375 + 3750 \sqrt{5})}{2 \cdot 1}$

Step 23.2.4.3.4.3

Rewrite the expression.

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 + \frac{- 9375 + 3750 \sqrt{5}}{1}$

Step 23.2.4.3.4.4

Divide $- 9375 + 3750 \sqrt{5}$ by $1$ .

$f (\frac{5 - \sqrt{5}}{2}) = 1875 - 825 \sqrt{5} + 7500 - 3000 \sqrt{5} - 1 - 9375 + 3750 \sqrt{5}$

Step 23.2.4.4

Simplify by adding and subtracting.

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Step 23.2.4.4.1

Add $1875$ and $7500$ .

$f (\frac{5 - \sqrt{5}}{2}) = 9375 - 825 \sqrt{5} - 3000 \sqrt{5} - 1 - 9375 + 3750 \sqrt{5}$

Step 23.2.4.4.2

Subtract $1$ from $9375$ .

$f (\frac{5 - \sqrt{5}}{2}) = 9374 - 825 \sqrt{5} - 3000 \sqrt{5} - 9375 + 3750 \sqrt{5}$

Step 23.2.4.4.3

Subtract $9375$ from $9374$ .

$f (\frac{5 - \sqrt{5}}{2}) = - 1 - 825 \sqrt{5} - 3000 \sqrt{5} + 3750 \sqrt{5}$

Step 23.2.4.5

Subtract $3000 \sqrt{5}$ from $- 825 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = - 1 - 3825 \sqrt{5} + 3750 \sqrt{5}$

Step 23.2.4.6

Add $- 3825 \sqrt{5}$ and $3750 \sqrt{5}$ .

$f (\frac{5 - \sqrt{5}}{2}) = - 1 - 75 \sqrt{5}$

Step 23.2.5

The final answer is $- 1 - 75 \sqrt{5}$ .

$y = - 1 - 75 \sqrt{5}$

Step 24

These are the local extrema for $f (x) = 6 x^{5} - 75 x^{4} + 300 x^{3} - 375 x^{2} - 1$ .

$(0, - 1)$ is a local maxima

$(5, - 1)$ is a local minima

$(\frac{5 + \sqrt{5}}{2}, - 1 + 75 \sqrt{5})$ is a local maxima

$(\frac{5 - \sqrt{5}}{2}, - 1 - 75 \sqrt{5})$ is a local minima

Step 25