Algebra Examples

$f (x) = x^{4} - 5 x^{3} + 3 x^{2} + 9 x - 3$ ; $(- 5, 5)$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 5 x^{3} + 3 x^{2} + 9 x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 5 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- 3]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 5 x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.1.2

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

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 5 x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- 5 x^{3}]$ .

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Step 1.1.1.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{3}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 4 x^{3} - 5 \frac{d}{d x} x^{3} + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.2.2

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

$f' (x) = 4 x^{3} - 5 (3 x^{2}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.2.3

Multiply $3$ by $- 5$ .

$f' (x) = 4 x^{3} - 15 x^{2} + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 4 x^{3} - 15 x^{2} + 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.3.2

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

$f' (x) = 4 x^{3} - 15 x^{2} + 3 (2 x) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.3.3

Multiply $2$ by $3$ .

$f' (x) = 4 x^{3} - 15 x^{2} + 6 x + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 3)$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 1.1.1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$f' (x) = 4 x^{3} - 15 x^{2} + 6 x + 9 \frac{d}{d x} (x) + \frac{d}{d x} (- 3)$

Step 1.1.1.4.2

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

$f' (x) = 4 x^{3} - 15 x^{2} + 6 x + 9 \cdot 1 + \frac{d}{d x} (- 3)$

Step 1.1.1.4.3

Multiply $9$ by $1$ .

$f' (x) = 4 x^{3} - 15 x^{2} + 6 x + 9 + \frac{d}{d x} (- 3)$

Step 1.1.1.5

Differentiate using the Constant Rule.

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Step 1.1.1.5.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} - 15 x^{2} + 6 x + 9 + 0$

Step 1.1.1.5.2

Add $4 x^{3} - 15 x^{2} + 6 x + 9$ and $0$ .

$f' (x) = 4 x^{3} - 15 x^{2} + 6 x + 9$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 15 x^{2} + 6 x + 9$ .

$4 x^{3} - 15 x^{2} + 6 x + 9$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 15 x^{2} + 6 x + 9 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 15 x^{2} + 6 x + 9 = 0$

Step 1.2.2

Factor $4 x^{3} - 15 x^{2} + 6 x + 9$ using the rational roots test.

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Step 1.2.2.1

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.

$p = \pm 1, \pm 9, \pm 3$

$q = \pm 1, \pm 4, \pm 2$

Step 1.2.2.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 9, \pm 2.25, \pm 4.5, \pm 3, \pm 0.75, \pm 1.5$

Step 1.2.2.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 1.2.2.3.1

Substitute $3$ into the polynomial.

$4 \cdot 3^{3} - 15 \cdot 3^{2} + 6 \cdot 3 + 9$

Step 1.2.2.3.2

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

$4 \cdot 27 - 15 \cdot 3^{2} + 6 \cdot 3 + 9$

Step 1.2.2.3.3

Multiply $4$ by $27$ .

$108 - 15 \cdot 3^{2} + 6 \cdot 3 + 9$

Step 1.2.2.3.4

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

$108 - 15 \cdot 9 + 6 \cdot 3 + 9$

Step 1.2.2.3.5

Multiply $- 15$ by $9$ .

$108 - 135 + 6 \cdot 3 + 9$

Step 1.2.2.3.6

Subtract $135$ from $108$ .

$- 27 + 6 \cdot 3 + 9$

Step 1.2.2.3.7

Multiply $6$ by $3$ .

$- 27 + 18 + 9$

Step 1.2.2.3.8

Add $- 27$ and $18$ .

$- 9 + 9$

Step 1.2.2.3.9

Add $- 9$ and $9$ .

$0$

Step 1.2.2.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 15 x^{2} + 6 x + 9}{x - 3}$

Step 1.2.2.5

Divide $4 x^{3} - 15 x^{2} + 6 x + 9$ by $x - 3$ .

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Step 1.2.2.5.1

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

$x$

$3$

$4 x^{3}$

$15 x^{2}$

$6 x$

$9$

Step 1.2.2.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $x$ .

					$4 x^{2}$
	$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$

Step 1.2.2.5.3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			+	$4 x^{3}$	-	$12 x^{2}$

Step 1.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 12 x^{2}$

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$

Step 1.2.2.5.5

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

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$

Step 1.2.2.5.6

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

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$

Step 1.2.2.5.7

Divide the highest order term in the dividend $- 3 x^{2}$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$

Step 1.2.2.5.8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					-	$3 x^{2}$	+	$9 x$

Step 1.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 9 x$

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$

Step 1.2.2.5.10

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

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$

Step 1.2.2.5.11

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

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$

Step 1.2.2.5.12

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$

Step 1.2.2.5.13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$
							-	$3 x$	+	$9$

Step 1.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x + 9$

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$
							+	$3 x$	-	$9$

Step 1.2.2.5.15

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

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$
							+	$3 x$	-	$9$
										$0$

Step 1.2.2.5.16

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

$4 x^{2} - 3 x - 3$

Step 1.2.2.6

Write $4 x^{3} - 15 x^{2} + 6 x + 9$ as a set of factors.

$(x - 3) (4 x^{2} - 3 x - 3) = 0$

Step 1.2.3

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

$x - 3 = 0$

$4 x^{2} - 3 x - 3 = 0$

Step 1.2.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.5

Set $4 x^{2} - 3 x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $4 x^{2} - 3 x - 3$ equal to $0$ .

$4 x^{2} - 3 x - 3 = 0$

Step 1.2.5.2

Solve $4 x^{2} - 3 x - 3 = 0$ for $x$ .

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Step 1.2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 1.2.5.2.2

Substitute the values $a = 4$ , $b = - 3$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (4 \cdot - 3)}}{2 \cdot 4}$

Step 1.2.5.2.3

Simplify.

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Step 1.2.5.2.3.1

Simplify the numerator.

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Step 1.2.5.2.3.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 1.2.5.2.3.1.2

Multiply $- 4 \cdot 4 \cdot - 3$ .

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 3}}{2 \cdot 4}$

Step 1.2.5.2.3.1.2.2

Multiply $- 16$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 48}}{2 \cdot 4}$

Step 1.2.5.2.3.1.3

Add $9$ and $48$ .

$x = \frac{3 \pm \sqrt{57}}{2 \cdot 4}$

Step 1.2.5.2.3.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{57}}{8}$

Step 1.2.5.2.4

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

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Step 1.2.5.2.4.1

Simplify the numerator.

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Step 1.2.5.2.4.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 1.2.5.2.4.1.2

Multiply $- 4 \cdot 4 \cdot - 3$ .

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 3}}{2 \cdot 4}$

Step 1.2.5.2.4.1.2.2

Multiply $- 16$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 48}}{2 \cdot 4}$

Step 1.2.5.2.4.1.3

Add $9$ and $48$ .

$x = \frac{3 \pm \sqrt{57}}{2 \cdot 4}$

Step 1.2.5.2.4.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{57}}{8}$

Step 1.2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{57}}{8}$

Step 1.2.5.2.5

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

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Step 1.2.5.2.5.1

Simplify the numerator.

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Step 1.2.5.2.5.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 1.2.5.2.5.1.2

Multiply $- 4 \cdot 4 \cdot - 3$ .

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 3}}{2 \cdot 4}$

Step 1.2.5.2.5.1.2.2

Multiply $- 16$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 48}}{2 \cdot 4}$

Step 1.2.5.2.5.1.3

Add $9$ and $48$ .

$x = \frac{3 \pm \sqrt{57}}{2 \cdot 4}$

Step 1.2.5.2.5.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{57}}{8}$

Step 1.2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{57}}{8}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{57}}{8}, \frac{3 - \sqrt{57}}{8}$

Step 1.2.6

The final solution is all the values that make $(x - 3) (4 x^{2} - 3 x - 3) = 0$ true.

$x = 3, \frac{3 + \sqrt{57}}{8}, \frac{3 - \sqrt{57}}{8}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $x^{4} - 5 x^{3} + 3 x^{2} + 9 x - 3$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 3$ .

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Step 1.4.1.1

Substitute $3$ for $x$ .

${(3)}^{4} - 5 {(3)}^{3} + 3 {(3)}^{2} + 9 (3) - 3$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Raise $3$ to the power of $4$ .

$81 - 5 {(3)}^{3} + 3 {(3)}^{2} + 9 (3) - 3$

Step 1.4.1.2.1.2

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

$81 - 5 \cdot 27 + 3 {(3)}^{2} + 9 (3) - 3$

Step 1.4.1.2.1.3

Multiply $- 5$ by $27$ .

$81 - 135 + 3 {(3)}^{2} + 9 (3) - 3$

Step 1.4.1.2.1.4

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 1.4.1.2.1.4.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 1.4.1.2.1.4.1.1

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

$81 - 135 + 3^{1} {(3)}^{2} + 9 (3) - 3$

Step 1.4.1.2.1.4.1.2

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

$81 - 135 + 3^{1 + 2} + 9 (3) - 3$

Step 1.4.1.2.1.4.2

Add $1$ and $2$ .

$81 - 135 + 3^{3} + 9 (3) - 3$

Step 1.4.1.2.1.5

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

$81 - 135 + 27 + 9 (3) - 3$

Step 1.4.1.2.1.6

Multiply $9$ by $3$ .

$81 - 135 + 27 + 27 - 3$

Step 1.4.1.2.2

Simplify by adding and subtracting.

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Step 1.4.1.2.2.1

Subtract $135$ from $81$ .

$- 54 + 27 + 27 - 3$

Step 1.4.1.2.2.2

Add $- 54$ and $27$ .

$- 27 + 27 - 3$

Step 1.4.1.2.2.3

Add $- 27$ and $27$ .

$0 - 3$

Step 1.4.1.2.2.4

Subtract $3$ from $0$ .

$- 3$

Step 1.4.2

Evaluate at $x = \frac{3 + \sqrt{57}}{8}$ .

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Step 1.4.2.1

Substitute $\frac{3 + \sqrt{57}}{8}$ for $x$ .

${(\frac{3 + \sqrt{57}}{8})}^{4} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

Apply the product rule to $\frac{3 + \sqrt{57}}{8}$ .

$\frac{{(3 + \sqrt{57})}^{4}}{8^{4}} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.2

Raise $8$ to the power of $4$ .

$\frac{{(3 + \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.3

Use the Binomial Theorem.

$\frac{3^{4} + 4 \cdot 3^{3} \sqrt{57} + 6 \cdot 3^{2} {\sqrt{57}}^{2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4

Simplify each term.

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Step 1.4.2.2.1.4.1

Raise $3$ to the power of $4$ .

$\frac{81 + 4 \cdot 3^{3} \sqrt{57} + 6 \cdot 3^{2} {\sqrt{57}}^{2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.2

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

$\frac{81 + 4 \cdot 27 \sqrt{57} + 6 \cdot 3^{2} {\sqrt{57}}^{2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.3

Multiply $4$ by $27$ .

$\frac{81 + 108 \sqrt{57} + 6 \cdot 3^{2} {\sqrt{57}}^{2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.4

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

$\frac{81 + 108 \sqrt{57} + 6 \cdot 9 {\sqrt{57}}^{2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.5

Multiply $6$ by $9$ .

$\frac{81 + 108 \sqrt{57} + 54 {\sqrt{57}}^{2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.6

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 1.4.2.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{81 + 108 \sqrt{57} + 54 {(57^{\frac{1}{2}})}^{2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.6.2

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

$\frac{81 + 108 \sqrt{57} + 54 \cdot 57^{\frac{1}{2} \cdot 2} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.6.3

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

$\frac{81 + 108 \sqrt{57} + 54 \cdot 57^{\frac{2}{2}} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.4.6.4.1

Cancel the common factor.

Step 1.4.2.2.1.4.6.4.2

Rewrite the expression.

$\frac{81 + 108 \sqrt{57} + 54 \cdot 57^{1} + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.6.5

Evaluate the exponent.

$\frac{81 + 108 \sqrt{57} + 54 \cdot 57 + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.7

Multiply $54$ by $57$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 4 \cdot 3 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.8

Multiply $4$ by $3$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 12 {\sqrt{57}}^{3} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.9

Rewrite ${\sqrt{57}}^{3}$ as $\sqrt{57^{3}}$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 12 \sqrt{57^{3}} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.10

Raise $57$ to the power of $3$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 12 \sqrt{185193} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.11

Rewrite $185193$ as $57^{2} \cdot 57$ .

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Step 1.4.2.2.1.4.11.1

Factor $3249$ out of $185193$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 12 \sqrt{3249 (57)} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.11.2

Rewrite $3249$ as $57^{2}$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 12 \sqrt{57^{2} \cdot 57} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.12

Pull terms out from under the radical.

$\frac{81 + 108 \sqrt{57} + 3078 + 12 (57 \sqrt{57}) + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.13

Multiply $57$ by $12$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14

Rewrite ${\sqrt{57}}^{4}$ as $57^{2}$ .

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Step 1.4.2.2.1.4.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + {(57^{\frac{1}{2}})}^{4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14.2

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

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 57^{\frac{1}{2} \cdot 4}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14.3

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

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 57^{\frac{4}{2}}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14.4

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

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Step 1.4.2.2.1.4.14.4.1

Factor $2$ out of $4$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 57^{\frac{2 \cdot 2}{2}}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14.4.2

Cancel the common factors.

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Step 1.4.2.2.1.4.14.4.2.1

Factor $2$ out of $2$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 57^{\frac{2 \cdot 2}{2 (1)}}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14.4.2.2

Cancel the common factor.

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 57^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14.4.2.3

Rewrite the expression.

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 57^{\frac{2}{1}}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.14.4.2.4

Divide $2$ by $1$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 57^{2}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.4.15

Raise $57$ to the power of $2$ .

$\frac{81 + 108 \sqrt{57} + 3078 + 684 \sqrt{57} + 3249}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.5

Add $81$ and $3078$ .

$\frac{3159 + 108 \sqrt{57} + 684 \sqrt{57} + 3249}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.6

Add $3159$ and $3249$ .

$\frac{6408 + 108 \sqrt{57} + 684 \sqrt{57}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.7

Add $108 \sqrt{57}$ and $684 \sqrt{57}$ .

$\frac{6408 + 792 \sqrt{57}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.8

Cancel the common factor of $6408 + 792 \sqrt{57}$ and $4096$ .

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Step 1.4.2.2.1.8.1

Factor $8$ out of $6408$ .

$\frac{8 (801) + 792 \sqrt{57}}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.8.2

Factor $8$ out of $792 \sqrt{57}$ .

$\frac{8 (801) + 8 (99 \sqrt{57})}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.8.3

Factor $8$ out of $8 (801) + 8 (99 \sqrt{57})$ .

$\frac{8 (801 + 99 \sqrt{57})}{4096} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.8.4

Cancel the common factors.

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Step 1.4.2.2.1.8.4.1

Factor $8$ out of $4096$ .

$\frac{8 (801 + 99 \sqrt{57})}{8 \cdot 512} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.8.4.2

Cancel the common factor.

$\frac{8 (801 + 99 \sqrt{57})}{8 \cdot 512} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.8.4.3

Rewrite the expression.

$\frac{801 + 99 \sqrt{57}}{512} - 5 {(\frac{3 + \sqrt{57}}{8})}^{3} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.9

Apply the product rule to $\frac{3 + \sqrt{57}}{8}$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{{(3 + \sqrt{57})}^{3}}{8^{3}} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.10

Raise $8$ to the power of $3$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{{(3 + \sqrt{57})}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.11

Use the Binomial Theorem.

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{3^{3} + 3 \cdot 3^{2} \sqrt{57} + 3 \cdot 3 {\sqrt{57}}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12

Simplify each term.

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Step 1.4.2.2.1.12.1

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

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 3 \cdot 3^{2} \sqrt{57} + 3 \cdot 3 {\sqrt{57}}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 1.4.2.2.1.12.2.1

Multiply $3$ by $3^{2}$ .

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Step 1.4.2.2.1.12.2.1.1

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

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 3^{1} \cdot 3^{2} \sqrt{57} + 3 \cdot 3 {\sqrt{57}}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.2.1.2

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

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 3^{1 + 2} \sqrt{57} + 3 \cdot 3 {\sqrt{57}}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.2.2

Add $1$ and $2$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 3^{3} \sqrt{57} + 3 \cdot 3 {\sqrt{57}}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.3

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

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 3 \cdot 3 {\sqrt{57}}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.4

Multiply $3$ by $3$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 9 {\sqrt{57}}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.5

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 1.4.2.2.1.12.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 9 {(57^{\frac{1}{2}})}^{2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.5.2

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

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 9 \cdot 57^{\frac{1}{2} \cdot 2} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.5.3

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

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 9 \cdot 57^{\frac{2}{2}} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.5.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.12.5.4.1

Cancel the common factor.

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 9 \cdot 57^{\frac{2}{2}} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.5.4.2

Rewrite the expression.

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 9 \cdot 57^{1} + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.5.5

Evaluate the exponent.

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 9 \cdot 57 + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.6

Multiply $9$ by $57$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 513 + {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.7

Rewrite ${\sqrt{57}}^{3}$ as $\sqrt{57^{3}}$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 513 + \sqrt{57^{3}}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.8

Raise $57$ to the power of $3$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 513 + \sqrt{185193}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.9

Rewrite $185193$ as $57^{2} \cdot 57$ .

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Step 1.4.2.2.1.12.9.1

Factor $3249$ out of $185193$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 513 + \sqrt{3249 (57)}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.9.2

Rewrite $3249$ as $57^{2}$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 513 + \sqrt{57^{2} \cdot 57}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.12.10

Pull terms out from under the radical.

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{27 + 27 \sqrt{57} + 513 + 57 \sqrt{57}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.13

Add $27$ and $513$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{540 + 27 \sqrt{57} + 57 \sqrt{57}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.14

Add $27 \sqrt{57}$ and $57 \sqrt{57}$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{540 + 84 \sqrt{57}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.15

Cancel the common factor of $540 + 84 \sqrt{57}$ and $512$ .

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Step 1.4.2.2.1.15.1

Factor $4$ out of $540$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{4 (135) + 84 \sqrt{57}}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.15.2

Factor $4$ out of $84 \sqrt{57}$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{4 (135) + 4 (21 \sqrt{57})}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.15.3

Factor $4$ out of $4 (135) + 4 (21 \sqrt{57})$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{4 (135 + 21 \sqrt{57})}{512} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.15.4

Cancel the common factors.

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Step 1.4.2.2.1.15.4.1

Factor $4$ out of $512$ .

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{4 (135 + 21 \sqrt{57})}{4 \cdot 128} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.15.4.2

Cancel the common factor.

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{4 (135 + 21 \sqrt{57})}{4 \cdot 128} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.15.4.3

Rewrite the expression.

$\frac{801 + 99 \sqrt{57}}{512} - 5 \frac{135 + 21 \sqrt{57}}{128} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.16

Combine $- 5$ and $\frac{135 + 21 \sqrt{57}}{128}$ .

$\frac{801 + 99 \sqrt{57}}{512} + \frac{- 5 (135 + 21 \sqrt{57})}{128} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.17

Move the negative in front of the fraction.

$\frac{801 + 99 \sqrt{57}}{512} - \frac{(5) (135 + 21 \sqrt{57})}{128} + 3 {(\frac{3 + \sqrt{57}}{8})}^{2} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.18

Apply the product rule to $\frac{3 + \sqrt{57}}{8}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{{(3 + \sqrt{57})}^{2}}{8^{2}} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.19

Raise $8$ to the power of $2$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{{(3 + \sqrt{57})}^{2}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.20

Rewrite ${(3 + \sqrt{57})}^{2}$ as $(3 + \sqrt{57}) (3 + \sqrt{57})$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{(3 + \sqrt{57}) (3 + \sqrt{57})}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.21

Expand $(3 + \sqrt{57}) (3 + \sqrt{57})$ using the FOIL Method.

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Step 1.4.2.2.1.21.1

Apply the distributive property.

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{3 (3 + \sqrt{57}) + \sqrt{57} (3 + \sqrt{57})}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.21.2

Apply the distributive property.

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{3 \cdot 3 + 3 \sqrt{57} + \sqrt{57} (3 + \sqrt{57})}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.21.3

Apply the distributive property.

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{3 \cdot 3 + 3 \sqrt{57} + \sqrt{57} \cdot 3 + \sqrt{57} \sqrt{57}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22

Simplify and combine like terms.

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Step 1.4.2.2.1.22.1

Simplify each term.

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Step 1.4.2.2.1.22.1.1

Multiply $3$ by $3$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{9 + 3 \sqrt{57} + \sqrt{57} \cdot 3 + \sqrt{57} \sqrt{57}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22.1.2

Move $3$ to the left of $\sqrt{57}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{9 + 3 \sqrt{57} + 3 \cdot \sqrt{57} + \sqrt{57} \sqrt{57}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22.1.3

Combine using the product rule for radicals.

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{9 + 3 \sqrt{57} + 3 \sqrt{57} + \sqrt{57 \cdot 57}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22.1.4

Multiply $57$ by $57$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{9 + 3 \sqrt{57} + 3 \sqrt{57} + \sqrt{3249}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22.1.5

Rewrite $3249$ as $57^{2}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{9 + 3 \sqrt{57} + 3 \sqrt{57} + \sqrt{57^{2}}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22.1.6

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

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{9 + 3 \sqrt{57} + 3 \sqrt{57} + 57}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22.2

Add $9$ and $57$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{66 + 3 \sqrt{57} + 3 \sqrt{57}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.22.3

Add $3 \sqrt{57}$ and $3 \sqrt{57}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{66 + 6 \sqrt{57}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.23

Cancel the common factor of $66 + 6 \sqrt{57}$ and $64$ .

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Step 1.4.2.2.1.23.1

Factor $2$ out of $66$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{2 (33) + 6 \sqrt{57}}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.23.2

Factor $2$ out of $6 \sqrt{57}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{2 (33) + 2 (3 \sqrt{57})}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.23.3

Factor $2$ out of $2 (33) + 2 (3 \sqrt{57})$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{2 (33 + 3 \sqrt{57})}{64} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.23.4

Cancel the common factors.

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Step 1.4.2.2.1.23.4.1

Factor $2$ out of $64$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{2 (33 + 3 \sqrt{57})}{2 \cdot 32} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.23.4.2

Cancel the common factor.

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{2 (33 + 3 \sqrt{57})}{2 \cdot 32} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.23.4.3

Rewrite the expression.

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + 3 \frac{33 + 3 \sqrt{57}}{32} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.24

Combine $3$ and $\frac{33 + 3 \sqrt{57}}{32}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + \frac{3 (33 + 3 \sqrt{57})}{32} + 9 (\frac{3 + \sqrt{57}}{8}) - 3$

Step 1.4.2.2.1.25

Combine $9$ and $\frac{3 + \sqrt{57}}{8}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57})}{128} + \frac{3 (33 + 3 \sqrt{57})}{32} + \frac{9 (3 + \sqrt{57})}{8} - 3$

Step 1.4.2.2.2

Find the common denominator.

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Step 1.4.2.2.2.1

Multiply $\frac{5 (135 + 21 \sqrt{57})}{128}$ by $\frac{4}{4}$ .

$\frac{801 + 99 \sqrt{57}}{512} - (\frac{5 (135 + 21 \sqrt{57})}{128} \cdot \frac{4}{4}) + \frac{3 (33 + 3 \sqrt{57})}{32} + \frac{9 (3 + \sqrt{57})}{8} - 3$

Step 1.4.2.2.2.2

Multiply $\frac{5 (135 + 21 \sqrt{57})}{128}$ by $\frac{4}{4}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 + 3 \sqrt{57})}{32} + \frac{9 (3 + \sqrt{57})}{8} - 3$

Step 1.4.2.2.2.3

Multiply $\frac{3 (33 + 3 \sqrt{57})}{32}$ by $\frac{16}{16}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 + 3 \sqrt{57})}{32} \cdot \frac{16}{16} + \frac{9 (3 + \sqrt{57})}{8} - 3$

Step 1.4.2.2.2.4

Multiply $\frac{3 (33 + 3 \sqrt{57})}{32}$ by $\frac{16}{16}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 + \sqrt{57})}{8} - 3$

Step 1.4.2.2.2.5

Multiply $\frac{9 (3 + \sqrt{57})}{8}$ by $\frac{64}{64}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 + \sqrt{57})}{8} \cdot \frac{64}{64} - 3$

Step 1.4.2.2.2.6

Multiply $\frac{9 (3 + \sqrt{57})}{8}$ by $\frac{64}{64}$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 + \sqrt{57}) \cdot 64}{8 \cdot 64} - 3$

Step 1.4.2.2.2.7

Write $- 3$ as a fraction with denominator $1$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 + \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3}{1}$

Step 1.4.2.2.2.8

Multiply $\frac{- 3}{1}$ by $\frac{512}{512}$ .

Step 1.4.2.2.2.9

Multiply $\frac{- 3}{1}$ by $\frac{512}{512}$ .

Step 1.4.2.2.2.10

Reorder the factors of $128 \cdot 4$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{4 \cdot 128} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 + \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.2.2.2.11

Multiply $4$ by $128$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 + \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.2.2.2.12

Reorder the factors of $32 \cdot 16$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{16 \cdot 32} + \frac{9 (3 + \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.2.2.2.13

Multiply $16$ by $32$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{512} + \frac{9 (3 + \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.2.2.2.14

Multiply $8$ by $64$ .

$\frac{801 + 99 \sqrt{57}}{512} - \frac{5 (135 + 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 + 3 \sqrt{57}) \cdot 16}{512} + \frac{9 (3 + \sqrt{57}) \cdot 64}{512} + \frac{- 3 \cdot 512}{512}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$\frac{801 + 99 \sqrt{57} - 5 (135 + 21 \sqrt{57}) \cdot 4 + 3 (33 + 3 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4

Simplify each term.

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Step 1.4.2.2.4.1

Apply the distributive property.

$\frac{801 + 99 \sqrt{57} + (- 5 \cdot 135 - 5 (21 \sqrt{57})) \cdot 4 + 3 (33 + 3 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.2

Multiply $- 5$ by $135$ .

$\frac{801 + 99 \sqrt{57} + (- 675 - 5 (21 \sqrt{57})) \cdot 4 + 3 (33 + 3 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.3

Multiply $21$ by $- 5$ .

$\frac{801 + 99 \sqrt{57} + (- 675 - 105 \sqrt{57}) \cdot 4 + 3 (33 + 3 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.4

Apply the distributive property.

$\frac{801 + 99 \sqrt{57} - 675 \cdot 4 - 105 \sqrt{57} \cdot 4 + 3 (33 + 3 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.5

Multiply $- 675$ by $4$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 105 \sqrt{57} \cdot 4 + 3 (33 + 3 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.6

Multiply $4$ by $- 105$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 3 (33 + 3 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.7

Apply the distributive property.

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + (3 \cdot 33 + 3 (3 \sqrt{57})) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.8

Multiply $3$ by $33$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + (99 + 3 (3 \sqrt{57})) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.9

Multiply $3$ by $3$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + (99 + 9 \sqrt{57}) \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.10

Apply the distributive property.

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 99 \cdot 16 + 9 \sqrt{57} \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.11

Multiply $99$ by $16$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 9 \sqrt{57} \cdot 16 + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.12

Multiply $16$ by $9$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + 9 (3 + \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.13

Apply the distributive property.

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + (9 \cdot 3 + 9 \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.14

Multiply $9$ by $3$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + (27 + 9 \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.15

Apply the distributive property.

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + 27 \cdot 64 + 9 \sqrt{57} \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.16

Multiply $27$ by $64$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + 1728 + 9 \sqrt{57} \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.2.2.4.17

Multiply $64$ by $9$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + 1728 + 576 \sqrt{57} - 3 \cdot 512}{512}$

Step 1.4.2.2.4.18

Multiply $- 3$ by $512$ .

$\frac{801 + 99 \sqrt{57} - 2700 - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + 1728 + 576 \sqrt{57} - 1536}{512}$

Step 1.4.2.2.5

Simplify terms.

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Step 1.4.2.2.5.1

Subtract $2700$ from $801$ .

$\frac{- 1899 + 99 \sqrt{57} - 420 \sqrt{57} + 1584 + 144 \sqrt{57} + 1728 + 576 \sqrt{57} - 1536}{512}$

Step 1.4.2.2.5.2

Simplify by adding and subtracting.

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Step 1.4.2.2.5.2.1

Add $- 1899$ and $1584$ .

$\frac{- 315 + 99 \sqrt{57} - 420 \sqrt{57} + 144 \sqrt{57} + 1728 + 576 \sqrt{57} - 1536}{512}$

Step 1.4.2.2.5.2.2

Add $- 315$ and $1728$ .

$\frac{1413 + 99 \sqrt{57} - 420 \sqrt{57} + 144 \sqrt{57} + 576 \sqrt{57} - 1536}{512}$

Step 1.4.2.2.5.2.3

Subtract $1536$ from $1413$ .

$\frac{- 123 + 99 \sqrt{57} - 420 \sqrt{57} + 144 \sqrt{57} + 576 \sqrt{57}}{512}$

Step 1.4.2.2.5.3

Subtract $420 \sqrt{57}$ from $99 \sqrt{57}$ .

$\frac{- 123 - 321 \sqrt{57} + 144 \sqrt{57} + 576 \sqrt{57}}{512}$

Step 1.4.2.2.5.4

Add $- 321 \sqrt{57}$ and $144 \sqrt{57}$ .

$\frac{- 123 - 177 \sqrt{57} + 576 \sqrt{57}}{512}$

Step 1.4.2.2.5.5

Add $- 177 \sqrt{57}$ and $576 \sqrt{57}$ .

$\frac{- 123 + 399 \sqrt{57}}{512}$

Step 1.4.2.2.5.6

Rewrite $- 123$ as $- 1 (123)$ .

$\frac{- 1 (123) + 399 \sqrt{57}}{512}$

Step 1.4.2.2.5.7

Factor $- 1$ out of $399 \sqrt{57}$ .

$\frac{- 1 (123) - (- 399 \sqrt{57})}{512}$

Step 1.4.2.2.5.8

Factor $- 1$ out of $- 1 (123) - (- 399 \sqrt{57})$ .

$\frac{- 1 (123 - 399 \sqrt{57})}{512}$

Step 1.4.2.2.5.9

Move the negative in front of the fraction.

$- \frac{123 - 399 \sqrt{57}}{512}$

Step 1.4.3

Evaluate at $x = \frac{3 - \sqrt{57}}{8}$ .

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Step 1.4.3.1

Substitute $\frac{3 - \sqrt{57}}{8}$ for $x$ .

${(\frac{3 - \sqrt{57}}{8})}^{4} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Simplify each term.

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Step 1.4.3.2.1.1

Apply the product rule to $\frac{3 - \sqrt{57}}{8}$ .

$\frac{{(3 - \sqrt{57})}^{4}}{8^{4}} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.2

Raise $8$ to the power of $4$ .

$\frac{{(3 - \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.3

Use the Binomial Theorem.

$\frac{3^{4} + 4 \cdot 3^{3} (- \sqrt{57}) + 6 \cdot 3^{2} {(- \sqrt{57})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4

Simplify each term.

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Step 1.4.3.2.1.4.1

Raise $3$ to the power of $4$ .

$\frac{81 + 4 \cdot 3^{3} (- \sqrt{57}) + 6 \cdot 3^{2} {(- \sqrt{57})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.2

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

$\frac{81 + 4 \cdot 27 (- \sqrt{57}) + 6 \cdot 3^{2} {(- \sqrt{57})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.3

Multiply $4$ by $27$ .

$\frac{81 + 108 (- \sqrt{57}) + 6 \cdot 3^{2} {(- \sqrt{57})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.4

Multiply $- 1$ by $108$ .

$\frac{81 - 108 \sqrt{57} + 6 \cdot 3^{2} {(- \sqrt{57})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.5

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

$\frac{81 - 108 \sqrt{57} + 6 \cdot 9 {(- \sqrt{57})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.6

Multiply $6$ by $9$ .

$\frac{81 - 108 \sqrt{57} + 54 {(- \sqrt{57})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.7

Apply the product rule to $- \sqrt{57}$ .

$\frac{81 - 108 \sqrt{57} + 54 ({(- 1)}^{2} {\sqrt{57}}^{2}) + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.8

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

$\frac{81 - 108 \sqrt{57} + 54 (1 {\sqrt{57}}^{2}) + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.9

Multiply ${\sqrt{57}}^{2}$ by $1$ .

$\frac{81 - 108 \sqrt{57} + 54 {\sqrt{57}}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.10

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 1.4.3.2.1.4.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{81 - 108 \sqrt{57} + 54 {(57^{\frac{1}{2}})}^{2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.10.2

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

$\frac{81 - 108 \sqrt{57} + 54 \cdot 57^{\frac{1}{2} \cdot 2} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.10.3

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

$\frac{81 - 108 \sqrt{57} + 54 \cdot 57^{\frac{2}{2}} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.10.4

Cancel the common factor of $2$ .

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Step 1.4.3.2.1.4.10.4.1

Cancel the common factor.

Step 1.4.3.2.1.4.10.4.2

Rewrite the expression.

$\frac{81 - 108 \sqrt{57} + 54 \cdot 57^{1} + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.10.5

Evaluate the exponent.

$\frac{81 - 108 \sqrt{57} + 54 \cdot 57 + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.11

Multiply $54$ by $57$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 4 \cdot 3 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.12

Multiply $4$ by $3$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 12 {(- \sqrt{57})}^{3} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.13

Apply the product rule to $- \sqrt{57}$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 12 ({(- 1)}^{3} {\sqrt{57}}^{3}) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.14

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

$\frac{81 - 108 \sqrt{57} + 3078 + 12 (- {\sqrt{57}}^{3}) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.15

Rewrite ${\sqrt{57}}^{3}$ as $\sqrt{57^{3}}$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 12 (- \sqrt{57^{3}}) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.16

Raise $57$ to the power of $3$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 12 (- \sqrt{185193}) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.17

Rewrite $185193$ as $57^{2} \cdot 57$ .

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Step 1.4.3.2.1.4.17.1

Factor $3249$ out of $185193$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 12 (- \sqrt{3249 (57)}) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.17.2

Rewrite $3249$ as $57^{2}$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 12 (- \sqrt{57^{2} \cdot 57}) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.18

Pull terms out from under the radical.

$\frac{81 - 108 \sqrt{57} + 3078 + 12 (- (57 \sqrt{57})) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.19

Multiply $57$ by $- 1$ .

$\frac{81 - 108 \sqrt{57} + 3078 + 12 (- 57 \sqrt{57}) + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.20

Multiply $- 57$ by $12$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + {(- \sqrt{57})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.21

Apply the product rule to $- \sqrt{57}$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + {(- 1)}^{4} {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.22

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

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 1 {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.23

Multiply ${\sqrt{57}}^{4}$ by $1$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + {\sqrt{57}}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24

Rewrite ${\sqrt{57}}^{4}$ as $57^{2}$ .

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Step 1.4.3.2.1.4.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + {(57^{\frac{1}{2}})}^{4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24.2

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

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 57^{\frac{1}{2} \cdot 4}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24.3

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

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 57^{\frac{4}{2}}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24.4

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

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Step 1.4.3.2.1.4.24.4.1

Factor $2$ out of $4$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 57^{\frac{2 \cdot 2}{2}}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24.4.2

Cancel the common factors.

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Step 1.4.3.2.1.4.24.4.2.1

Factor $2$ out of $2$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 57^{\frac{2 \cdot 2}{2 (1)}}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24.4.2.2

Cancel the common factor.

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 57^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24.4.2.3

Rewrite the expression.

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 57^{\frac{2}{1}}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.24.4.2.4

Divide $2$ by $1$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 57^{2}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.4.25

Raise $57$ to the power of $2$ .

$\frac{81 - 108 \sqrt{57} + 3078 - 684 \sqrt{57} + 3249}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.5

Add $81$ and $3078$ .

$\frac{3159 - 108 \sqrt{57} - 684 \sqrt{57} + 3249}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.6

Add $3159$ and $3249$ .

$\frac{6408 - 108 \sqrt{57} - 684 \sqrt{57}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.7

Subtract $684 \sqrt{57}$ from $- 108 \sqrt{57}$ .

$\frac{6408 - 792 \sqrt{57}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.8

Cancel the common factor of $6408 - 792 \sqrt{57}$ and $4096$ .

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Step 1.4.3.2.1.8.1

Factor $8$ out of $6408$ .

$\frac{8 (801) - 792 \sqrt{57}}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.8.2

Factor $8$ out of $- 792 \sqrt{57}$ .

$\frac{8 (801) + 8 (- 99 \sqrt{57})}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.8.3

Factor $8$ out of $8 (801) + 8 (- 99 \sqrt{57})$ .

$\frac{8 (801 - 99 \sqrt{57})}{4096} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.8.4

Cancel the common factors.

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Step 1.4.3.2.1.8.4.1

Factor $8$ out of $4096$ .

$\frac{8 (801 - 99 \sqrt{57})}{8 \cdot 512} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.8.4.2

Cancel the common factor.

$\frac{8 (801 - 99 \sqrt{57})}{8 \cdot 512} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.8.4.3

Rewrite the expression.

$\frac{801 - 99 \sqrt{57}}{512} - 5 {(\frac{3 - \sqrt{57}}{8})}^{3} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.9

Apply the product rule to $\frac{3 - \sqrt{57}}{8}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{{(3 - \sqrt{57})}^{3}}{8^{3}} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.10

Raise $8$ to the power of $3$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{{(3 - \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.11

Use the Binomial Theorem.

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{3^{3} + 3 \cdot 3^{2} (- \sqrt{57}) + 3 \cdot 3 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12

Simplify each term.

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Step 1.4.3.2.1.12.1

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 + 3 \cdot 3^{2} (- \sqrt{57}) + 3 \cdot 3 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 1.4.3.2.1.12.2.1

Multiply $3$ by $3^{2}$ .

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Step 1.4.3.2.1.12.2.1.1

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 + 3^{1} \cdot 3^{2} (- \sqrt{57}) + 3 \cdot 3 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.2.1.2

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 + 3^{1 + 2} (- \sqrt{57}) + 3 \cdot 3 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.2.2

Add $1$ and $2$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 + 3^{3} (- \sqrt{57}) + 3 \cdot 3 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.3

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 + 27 (- \sqrt{57}) + 3 \cdot 3 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.4

Multiply $- 1$ by $27$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 3 \cdot 3 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.5

Multiply $3$ by $3$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 {(- \sqrt{57})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.6

Apply the product rule to $- \sqrt{57}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 ({(- 1)}^{2} {\sqrt{57}}^{2}) + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.7

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 (1 {\sqrt{57}}^{2}) + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.8

Multiply ${\sqrt{57}}^{2}$ by $1$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 {\sqrt{57}}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.9

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 1.4.3.2.1.12.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 {(57^{\frac{1}{2}})}^{2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.9.2

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 \cdot 57^{\frac{1}{2} \cdot 2} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.9.3

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 \cdot 57^{\frac{2}{2}} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.9.4

Cancel the common factor of $2$ .

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Step 1.4.3.2.1.12.9.4.1

Cancel the common factor.

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 \cdot 57^{\frac{2}{2}} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.9.4.2

Rewrite the expression.

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 \cdot 57^{1} + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.9.5

Evaluate the exponent.

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 9 \cdot 57 + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.10

Multiply $9$ by $57$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 + {(- \sqrt{57})}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.11

Apply the product rule to $- \sqrt{57}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 + {(- 1)}^{3} {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.12

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

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 - {\sqrt{57}}^{3}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.13

Rewrite ${\sqrt{57}}^{3}$ as $\sqrt{57^{3}}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 - \sqrt{57^{3}}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.14

Raise $57$ to the power of $3$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 - \sqrt{185193}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.15

Rewrite $185193$ as $57^{2} \cdot 57$ .

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Step 1.4.3.2.1.12.15.1

Factor $3249$ out of $185193$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 - \sqrt{3249 (57)}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.15.2

Rewrite $3249$ as $57^{2}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 - \sqrt{57^{2} \cdot 57}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.16

Pull terms out from under the radical.

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 - (57 \sqrt{57})}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.12.17

Multiply $57$ by $- 1$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{27 - 27 \sqrt{57} + 513 - 57 \sqrt{57}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.13

Add $27$ and $513$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{540 - 27 \sqrt{57} - 57 \sqrt{57}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.14

Subtract $57 \sqrt{57}$ from $- 27 \sqrt{57}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{540 - 84 \sqrt{57}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.15

Cancel the common factor of $540 - 84 \sqrt{57}$ and $512$ .

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Step 1.4.3.2.1.15.1

Factor $4$ out of $540$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{4 (135) - 84 \sqrt{57}}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.15.2

Factor $4$ out of $- 84 \sqrt{57}$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{4 (135) + 4 (- 21 \sqrt{57})}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.15.3

Factor $4$ out of $4 (135) + 4 (- 21 \sqrt{57})$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{4 (135 - 21 \sqrt{57})}{512} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.15.4

Cancel the common factors.

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Step 1.4.3.2.1.15.4.1

Factor $4$ out of $512$ .

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{4 (135 - 21 \sqrt{57})}{4 \cdot 128} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.15.4.2

Cancel the common factor.

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{4 (135 - 21 \sqrt{57})}{4 \cdot 128} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.15.4.3

Rewrite the expression.

$\frac{801 - 99 \sqrt{57}}{512} - 5 \frac{135 - 21 \sqrt{57}}{128} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.16

Combine $- 5$ and $\frac{135 - 21 \sqrt{57}}{128}$ .

$\frac{801 - 99 \sqrt{57}}{512} + \frac{- 5 (135 - 21 \sqrt{57})}{128} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.17

Move the negative in front of the fraction.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{(5) (135 - 21 \sqrt{57})}{128} + 3 {(\frac{3 - \sqrt{57}}{8})}^{2} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.18

Apply the product rule to $\frac{3 - \sqrt{57}}{8}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{{(3 - \sqrt{57})}^{2}}{8^{2}} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.19

Raise $8$ to the power of $2$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{{(3 - \sqrt{57})}^{2}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.20

Rewrite ${(3 - \sqrt{57})}^{2}$ as $(3 - \sqrt{57}) (3 - \sqrt{57})$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{(3 - \sqrt{57}) (3 - \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.21

Expand $(3 - \sqrt{57}) (3 - \sqrt{57})$ using the FOIL Method.

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Step 1.4.3.2.1.21.1

Apply the distributive property.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{3 (3 - \sqrt{57}) - \sqrt{57} (3 - \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.21.2

Apply the distributive property.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{3 \cdot 3 + 3 (- \sqrt{57}) - \sqrt{57} (3 - \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.21.3

Apply the distributive property.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{3 \cdot 3 + 3 (- \sqrt{57}) - \sqrt{57} \cdot 3 - \sqrt{57} (- \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22

Simplify and combine like terms.

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Step 1.4.3.2.1.22.1

Simplify each term.

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Step 1.4.3.2.1.22.1.1

Multiply $3$ by $3$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 + 3 (- \sqrt{57}) - \sqrt{57} \cdot 3 - \sqrt{57} (- \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.2

Multiply $- 1$ by $3$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - \sqrt{57} \cdot 3 - \sqrt{57} (- \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.3

Multiply $3$ by $- 1$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} - \sqrt{57} (- \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.4

Multiply $- \sqrt{57} (- \sqrt{57})$ .

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Step 1.4.3.2.1.22.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + 1 \sqrt{57} \sqrt{57}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.4.2

Multiply $\sqrt{57}$ by $1$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + \sqrt{57} \sqrt{57}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.4.3

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + {\sqrt{57}}^{1} \sqrt{57}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.4.4

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + {\sqrt{57}}^{1} {\sqrt{57}}^{1}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.4.5

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

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + {\sqrt{57}}^{1 + 1}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.4.6

Add $1$ and $1$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + {\sqrt{57}}^{2}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.5

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 1.4.3.2.1.22.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + {(57^{\frac{1}{2}})}^{2}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.5.2

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

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + 57^{\frac{1}{2} \cdot 2}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.5.3

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

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + 57^{\frac{2}{2}}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.5.4

Cancel the common factor of $2$ .

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Step 1.4.3.2.1.22.1.5.4.1

Cancel the common factor.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + 57^{\frac{2}{2}}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.5.4.2

Rewrite the expression.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + 57^{1}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.1.5.5

Evaluate the exponent.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{9 - 3 \sqrt{57} - 3 \sqrt{57} + 57}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.2

Add $9$ and $57$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{66 - 3 \sqrt{57} - 3 \sqrt{57}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.22.3

Subtract $3 \sqrt{57}$ from $- 3 \sqrt{57}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{66 - 6 \sqrt{57}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.23

Cancel the common factor of $66 - 6 \sqrt{57}$ and $64$ .

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Step 1.4.3.2.1.23.1

Factor $2$ out of $66$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{2 (33) - 6 \sqrt{57}}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.23.2

Factor $2$ out of $- 6 \sqrt{57}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{2 (33) + 2 (- 3 \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.23.3

Factor $2$ out of $2 (33) + 2 (- 3 \sqrt{57})$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{2 (33 - 3 \sqrt{57})}{64} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.23.4

Cancel the common factors.

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Step 1.4.3.2.1.23.4.1

Factor $2$ out of $64$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{2 (33 - 3 \sqrt{57})}{2 \cdot 32} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.23.4.2

Cancel the common factor.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{2 (33 - 3 \sqrt{57})}{2 \cdot 32} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.23.4.3

Rewrite the expression.

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + 3 \frac{33 - 3 \sqrt{57}}{32} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.24

Combine $3$ and $\frac{33 - 3 \sqrt{57}}{32}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + \frac{3 (33 - 3 \sqrt{57})}{32} + 9 (\frac{3 - \sqrt{57}}{8}) - 3$

Step 1.4.3.2.1.25

Combine $9$ and $\frac{3 - \sqrt{57}}{8}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57})}{128} + \frac{3 (33 - 3 \sqrt{57})}{32} + \frac{9 (3 - \sqrt{57})}{8} - 3$

Step 1.4.3.2.2

Find the common denominator.

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Step 1.4.3.2.2.1

Multiply $\frac{5 (135 - 21 \sqrt{57})}{128}$ by $\frac{4}{4}$ .

$\frac{801 - 99 \sqrt{57}}{512} - (\frac{5 (135 - 21 \sqrt{57})}{128} \cdot \frac{4}{4}) + \frac{3 (33 - 3 \sqrt{57})}{32} + \frac{9 (3 - \sqrt{57})}{8} - 3$

Step 1.4.3.2.2.2

Multiply $\frac{5 (135 - 21 \sqrt{57})}{128}$ by $\frac{4}{4}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 - 3 \sqrt{57})}{32} + \frac{9 (3 - \sqrt{57})}{8} - 3$

Step 1.4.3.2.2.3

Multiply $\frac{3 (33 - 3 \sqrt{57})}{32}$ by $\frac{16}{16}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 - 3 \sqrt{57})}{32} \cdot \frac{16}{16} + \frac{9 (3 - \sqrt{57})}{8} - 3$

Step 1.4.3.2.2.4

Multiply $\frac{3 (33 - 3 \sqrt{57})}{32}$ by $\frac{16}{16}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 - \sqrt{57})}{8} - 3$

Step 1.4.3.2.2.5

Multiply $\frac{9 (3 - \sqrt{57})}{8}$ by $\frac{64}{64}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 - \sqrt{57})}{8} \cdot \frac{64}{64} - 3$

Step 1.4.3.2.2.6

Multiply $\frac{9 (3 - \sqrt{57})}{8}$ by $\frac{64}{64}$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 - \sqrt{57}) \cdot 64}{8 \cdot 64} - 3$

Step 1.4.3.2.2.7

Write $- 3$ as a fraction with denominator $1$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{128 \cdot 4} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 - \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3}{1}$

Step 1.4.3.2.2.8

Multiply $\frac{- 3}{1}$ by $\frac{512}{512}$ .

Step 1.4.3.2.2.9

Multiply $\frac{- 3}{1}$ by $\frac{512}{512}$ .

Step 1.4.3.2.2.10

Reorder the factors of $128 \cdot 4$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{4 \cdot 128} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 - \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.3.2.2.11

Multiply $4$ by $128$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{32 \cdot 16} + \frac{9 (3 - \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.3.2.2.12

Reorder the factors of $32 \cdot 16$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{16 \cdot 32} + \frac{9 (3 - \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.3.2.2.13

Multiply $16$ by $32$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{512} + \frac{9 (3 - \sqrt{57}) \cdot 64}{8 \cdot 64} + \frac{- 3 \cdot 512}{512}$

Step 1.4.3.2.2.14

Multiply $8$ by $64$ .

$\frac{801 - 99 \sqrt{57}}{512} - \frac{5 (135 - 21 \sqrt{57}) \cdot 4}{512} + \frac{3 (33 - 3 \sqrt{57}) \cdot 16}{512} + \frac{9 (3 - \sqrt{57}) \cdot 64}{512} + \frac{- 3 \cdot 512}{512}$

Step 1.4.3.2.3

Combine the numerators over the common denominator.

$\frac{801 - 99 \sqrt{57} - 5 (135 - 21 \sqrt{57}) \cdot 4 + 3 (33 - 3 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4

Simplify each term.

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Step 1.4.3.2.4.1

Apply the distributive property.

$\frac{801 - 99 \sqrt{57} + (- 5 \cdot 135 - 5 (- 21 \sqrt{57})) \cdot 4 + 3 (33 - 3 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.2

Multiply $- 5$ by $135$ .

$\frac{801 - 99 \sqrt{57} + (- 675 - 5 (- 21 \sqrt{57})) \cdot 4 + 3 (33 - 3 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.3

Multiply $- 21$ by $- 5$ .

$\frac{801 - 99 \sqrt{57} + (- 675 + 105 \sqrt{57}) \cdot 4 + 3 (33 - 3 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.4

Apply the distributive property.

$\frac{801 - 99 \sqrt{57} - 675 \cdot 4 + 105 \sqrt{57} \cdot 4 + 3 (33 - 3 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.5

Multiply $- 675$ by $4$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 105 \sqrt{57} \cdot 4 + 3 (33 - 3 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.6

Multiply $4$ by $105$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 3 (33 - 3 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.7

Apply the distributive property.

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + (3 \cdot 33 + 3 (- 3 \sqrt{57})) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.8

Multiply $3$ by $33$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + (99 + 3 (- 3 \sqrt{57})) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.9

Multiply $- 3$ by $3$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + (99 - 9 \sqrt{57}) \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.10

Apply the distributive property.

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 99 \cdot 16 - 9 \sqrt{57} \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.11

Multiply $99$ by $16$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 9 \sqrt{57} \cdot 16 + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.12

Multiply $16$ by $- 9$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + 9 (3 - \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.13

Apply the distributive property.

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + (9 \cdot 3 + 9 (- \sqrt{57})) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.14

Multiply $9$ by $3$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + (27 + 9 (- \sqrt{57})) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.15

Multiply $- 1$ by $9$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + (27 - 9 \sqrt{57}) \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.16

Apply the distributive property.

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + 27 \cdot 64 - 9 \sqrt{57} \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.17

Multiply $27$ by $64$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + 1728 - 9 \sqrt{57} \cdot 64 - 3 \cdot 512}{512}$

Step 1.4.3.2.4.18

Multiply $64$ by $- 9$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + 1728 - 576 \sqrt{57} - 3 \cdot 512}{512}$

Step 1.4.3.2.4.19

Multiply $- 3$ by $512$ .

$\frac{801 - 99 \sqrt{57} - 2700 + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + 1728 - 576 \sqrt{57} - 1536}{512}$

Step 1.4.3.2.5

Simplify terms.

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Step 1.4.3.2.5.1

Subtract $2700$ from $801$ .

$\frac{- 1899 - 99 \sqrt{57} + 420 \sqrt{57} + 1584 - 144 \sqrt{57} + 1728 - 576 \sqrt{57} - 1536}{512}$

Step 1.4.3.2.5.2

Simplify by adding and subtracting.

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Step 1.4.3.2.5.2.1

Add $- 1899$ and $1584$ .

$\frac{- 315 - 99 \sqrt{57} + 420 \sqrt{57} - 144 \sqrt{57} + 1728 - 576 \sqrt{57} - 1536}{512}$

Step 1.4.3.2.5.2.2

Add $- 315$ and $1728$ .

$\frac{1413 - 99 \sqrt{57} + 420 \sqrt{57} - 144 \sqrt{57} - 576 \sqrt{57} - 1536}{512}$

Step 1.4.3.2.5.2.3

Subtract $1536$ from $1413$ .

$\frac{- 123 - 99 \sqrt{57} + 420 \sqrt{57} - 144 \sqrt{57} - 576 \sqrt{57}}{512}$

Step 1.4.3.2.5.3

Add $- 99 \sqrt{57}$ and $420 \sqrt{57}$ .

$\frac{- 123 + 321 \sqrt{57} - 144 \sqrt{57} - 576 \sqrt{57}}{512}$

Step 1.4.3.2.5.4

Subtract $144 \sqrt{57}$ from $321 \sqrt{57}$ .

$\frac{- 123 + 177 \sqrt{57} - 576 \sqrt{57}}{512}$

Step 1.4.3.2.5.5

Subtract $576 \sqrt{57}$ from $177 \sqrt{57}$ .

$\frac{- 123 - 399 \sqrt{57}}{512}$

Step 1.4.3.2.5.6

Rewrite $- 123$ as $- 1 (123)$ .

$\frac{- 1 (123) - 399 \sqrt{57}}{512}$

Step 1.4.3.2.5.7

Factor $- 1$ out of $- 399 \sqrt{57}$ .

$\frac{- 1 (123) - (399 \sqrt{57})}{512}$

Step 1.4.3.2.5.8

Factor $- 1$ out of $- 1 (123) - (399 \sqrt{57})$ .

$\frac{- 1 (123 + 399 \sqrt{57})}{512}$

Step 1.4.3.2.5.9

Move the negative in front of the fraction.

$- \frac{123 + 399 \sqrt{57}}{512}$

Step 1.4.4

List all of the points.

$(3, - 3), (\frac{3 + \sqrt{57}}{8}, - \frac{123 - 399 \sqrt{57}}{512}), (\frac{3 - \sqrt{57}}{8}, - \frac{123 + 399 \sqrt{57}}{512})$

Step 2

Use the first derivative test to determine which points can be maxima or minima.

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Step 2.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, \frac{3 - \sqrt{57}}{8}) \cup (\frac{3 - \sqrt{57}}{8}, \frac{3 + \sqrt{57}}{8}) \cup (\frac{3 + \sqrt{57}}{8}, 3) \cup (3, \infty)$

Step 2.2

Substitute any number, such as $- 3$ , from the interval $(- \infty, \frac{3 - \sqrt{57}}{8})$ in the first derivative $4 x^{3} - 15 x^{2} + 6 x + 9$ to check if the result is negative or positive.

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Step 2.2.1

Replace the variable $x$ with $- 3$ in the expression.

$f' (- 3) = 4 {(- 3)}^{3} - 15 {(- 3)}^{2} + 6 (- 3) + 9$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$f' (- 3) = 4 \cdot - 27 - 15 {(- 3)}^{2} + 6 (- 3) + 9$

Step 2.2.2.1.2

Multiply $4$ by $- 27$ .

$f' (- 3) = - 108 - 15 {(- 3)}^{2} + 6 (- 3) + 9$

Step 2.2.2.1.3

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

$f' (- 3) = - 108 - 15 \cdot 9 + 6 (- 3) + 9$

Step 2.2.2.1.4

Multiply $- 15$ by $9$ .

$f' (- 3) = - 108 - 135 + 6 (- 3) + 9$

Step 2.2.2.1.5

Multiply $6$ by $- 3$ .

$f' (- 3) = - 108 - 135 - 18 + 9$

Step 2.2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.2.1

Subtract $135$ from $- 108$ .

$f' (- 3) = - 243 - 18 + 9$

Step 2.2.2.2.2

Subtract $18$ from $- 243$ .

$f' (- 3) = - 261 + 9$

Step 2.2.2.2.3

Add $- 261$ and $9$ .

$f' (- 3) = - 252$

Step 2.2.2.3

The final answer is $- 252$ .

$- 252$

Step 2.3

Substitute any number, such as $0$ , from the interval $(\frac{3 - \sqrt{57}}{8}, \frac{3 + \sqrt{57}}{8})$ in the first derivative $4 x^{3} - 15 x^{2} + 6 x + 9$ to check if the result is negative or positive.

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Step 2.3.1

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

$f' (0) = 4 {(0)}^{3} - 15 {(0)}^{2} + 6 (0) + 9$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

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

$f' (0) = 4 \cdot 0 - 15 {(0)}^{2} + 6 (0) + 9$

Step 2.3.2.1.2

Multiply $4$ by $0$ .

$f' (0) = 0 - 15 {(0)}^{2} + 6 (0) + 9$

Step 2.3.2.1.3

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

$f' (0) = 0 - 15 \cdot 0 + 6 (0) + 9$

Step 2.3.2.1.4

Multiply $- 15$ by $0$ .

$f' (0) = 0 + 0 + 6 (0) + 9$

Step 2.3.2.1.5

Multiply $6$ by $0$ .

$f' (0) = 0 + 0 + 0 + 9$

Step 2.3.2.2

Simplify by adding numbers.

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Step 2.3.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 + 0 + 9$

Step 2.3.2.2.2

Add $0$ and $0$ .

$f' (0) = 0 + 9$

Step 2.3.2.2.3

Add $0$ and $9$ .

$f' (0) = 9$

Step 2.3.2.3

The final answer is $9$ .

$9$

Step 2.4

Substitute any number, such as $2$ , from the interval $(\frac{3 + \sqrt{57}}{8}, 3)$ in the first derivative $4 x^{3} - 15 x^{2} + 6 x + 9$ to check if the result is negative or positive.

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Step 2.4.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = 4 {(2)}^{3} - 15 {(2)}^{2} + 6 (2) + 9$

Step 2.4.2

Simplify the result.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

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

$f' (2) = 4 \cdot 8 - 15 {(2)}^{2} + 6 (2) + 9$

Step 2.4.2.1.2

Multiply $4$ by $8$ .

$f' (2) = 32 - 15 {(2)}^{2} + 6 (2) + 9$

Step 2.4.2.1.3

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

$f' (2) = 32 - 15 \cdot 4 + 6 (2) + 9$

Step 2.4.2.1.4

Multiply $- 15$ by $4$ .

$f' (2) = 32 - 60 + 6 (2) + 9$

Step 2.4.2.1.5

Multiply $6$ by $2$ .

$f' (2) = 32 - 60 + 12 + 9$

Step 2.4.2.2

Simplify by adding and subtracting.

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Step 2.4.2.2.1

Subtract $60$ from $32$ .

$f' (2) = - 28 + 12 + 9$

Step 2.4.2.2.2

Add $- 28$ and $12$ .

$f' (2) = - 16 + 9$

Step 2.4.2.2.3

Add $- 16$ and $9$ .

$f' (2) = - 7$

Step 2.4.2.3

The final answer is $- 7$ .

$- 7$

Step 2.5

Substitute any number, such as $6$ , from the interval $(3, \infty)$ in the first derivative $4 x^{3} - 15 x^{2} + 6 x + 9$ to check if the result is negative or positive.

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Step 2.5.1

Replace the variable $x$ with $6$ in the expression.

$f' (6) = 4 {(6)}^{3} - 15 {(6)}^{2} + 6 (6) + 9$

Step 2.5.2

Simplify the result.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Raise $6$ to the power of $3$ .

$f' (6) = 4 \cdot 216 - 15 {(6)}^{2} + 6 (6) + 9$

Step 2.5.2.1.2

Multiply $4$ by $216$ .

$f' (6) = 864 - 15 {(6)}^{2} + 6 (6) + 9$

Step 2.5.2.1.3

Raise $6$ to the power of $2$ .

$f' (6) = 864 - 15 \cdot 36 + 6 (6) + 9$

Step 2.5.2.1.4

Multiply $- 15$ by $36$ .

$f' (6) = 864 - 540 + 6 (6) + 9$

Step 2.5.2.1.5

Multiply $6$ by $6$ .

$f' (6) = 864 - 540 + 36 + 9$

Step 2.5.2.2

Simplify by adding and subtracting.

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Step 2.5.2.2.1

Subtract $540$ from $864$ .

$f' (6) = 324 + 36 + 9$

Step 2.5.2.2.2

Add $324$ and $36$ .

$f' (6) = 360 + 9$

Step 2.5.2.2.3

Add $360$ and $9$ .

$f' (6) = 369$

Step 2.5.2.3

The final answer is $369$ .

$369$

Step 2.6

Since the first derivative changed signs from negative to positive around $x = \frac{3 - \sqrt{57}}{8}$ , then $x = \frac{3 - \sqrt{57}}{8}$ is a local minimum.

$x = \frac{3 - \sqrt{57}}{8}$ is a local minimum

Step 2.7

Since the first derivative changed signs from positive to negative around $x = \frac{3 + \sqrt{57}}{8}$ , then $x = \frac{3 + \sqrt{57}}{8}$ is a local maximum.

$x = \frac{3 + \sqrt{57}}{8}$ is a local maximum

Step 2.8

Since the first derivative changed signs from negative to positive around $x = 3$ , then $x = 3$ is a local minimum.

$x = 3$ is a local minimum

Step 2.9

These are the local extrema for $f (x) = x^{4} - 5 x^{3} + 3 x^{2} + 9 x - 3$ .

$x = \frac{3 - \sqrt{57}}{8}$ is a local minimum

$x = \frac{3 + \sqrt{57}}{8}$ is a local maximum

$x = 3$ is a local minimum

$x = \frac{3 - \sqrt{57}}{8}$ is a local minimum

$x = \frac{3 + \sqrt{57}}{8}$ is a local maximum

$x = 3$ is a local minimum

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

No absolute maximum

Absolute Minimum: $(\frac{3 - \sqrt{57}}{8}, - \frac{123 + 399 \sqrt{57}}{512})$

Step 4