Calculus Examples

$f (x) = x^{4} - 2 x^{3} + x + 1$ , $[- 1, 3]$

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 $x^{4} - 2 x^{3} + x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

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 $n = 4$ .

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

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 $n = 3$ .

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

Step 1.1.1.2.3

Multiply $3$ by $- 2$ .

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

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

$f' (x) = 4 x^{3} - 6 x^{2} + 1 + 0$

Step 1.1.1.3.3

Add $4 x^{3} - 6 x^{2} + 1$ and $0$ .

$f' (x) = 4 x^{3} - 6 x^{2} + 1$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 6 x^{2} + 1$ .

$4 x^{3} - 6 x^{2} + 1$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 6 x^{2} + 1 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 6 x^{2} + 1 = 0$

Step 1.2.2

Factor $4 x^{3} - 6 x^{2} + 1$ 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.

$p = \pm 1$

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

Step 1.2.2.2

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

$\pm 1, \pm 0.25, \pm 0.5$

Step 1.2.2.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 1.2.2.3.1

Substitute $0.5$ into the polynomial.

$4 \cdot {0.5}^{3} - 6 \cdot {0.5}^{2} + 1$

Step 1.2.2.3.2

Raise $0.5$ to the power of $3$ .

$4 \cdot 0.125 - 6 \cdot {0.5}^{2} + 1$

Step 1.2.2.3.3

Multiply $4$ by $0.125$ .

$0.5 - 6 \cdot {0.5}^{2} + 1$

Step 1.2.2.3.4

Raise $0.5$ to the power of $2$ .

$0.5 - 6 \cdot 0.25 + 1$

Step 1.2.2.3.5

Multiply $- 6$ by $0.25$ .

$0.5 - 1.5 + 1$

Step 1.2.2.3.6

Subtract $1.5$ from $0.5$ .

$- 1 + 1$

Step 1.2.2.3.7

Add $- 1$ and $1$ .

$0$

Step 1.2.2.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 6 x^{2} + 1}{2 x - 1}$

Step 1.2.2.5

Divide $4 x^{3} - 6 x^{2} + 1$ by $2 x - 1$ .

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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 $0$ .

$2 x$

$1$

$4 x^{3}$

$6 x^{2}$

$0 x$

$1$

Step 1.2.2.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$

Step 1.2.2.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			+	$4 x^{3}$	-	$2 x^{2}$

Step 1.2.2.5.4

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$

Step 1.2.2.5.5

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$

Step 1.2.2.5.6

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 1.2.2.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$2 x$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 1.2.2.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$2 x$

Step 1.2.2.5.9

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

				$2 x^{2}$	-	$2 x$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$2 x$

Step 1.2.2.5.10

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

				$2 x^{2}$	-	$2 x$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$2 x$
							-	$2 x$

Step 1.2.2.5.11

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

				$2 x^{2}$	-	$2 x$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$2 x$
							-	$2 x$	+	$1$

Step 1.2.2.5.12

Divide the highest order term in the dividend $- 2 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$2 x$	-	$1$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$2 x$
							-	$2 x$	+	$1$

Step 1.2.2.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$	-	$1$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$2 x$
							-	$2 x$	+	$1$
							-	$2 x$	+	$1$

Step 1.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x + 1$

				$2 x^{2}$	-	$2 x$	-	$1$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$2 x$
							-	$2 x$	+	$1$
							+	$2 x$	-	$1$

Step 1.2.2.5.15

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

				$2 x^{2}$	-	$2 x$	-	$1$
$2 x$	-	$1$		$4 x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$1$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$2 x$
							-	$2 x$	+	$1$
							+	$2 x$	-	$1$
										$0$

Step 1.2.2.5.16

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

$2 x^{2} - 2 x - 1$

Step 1.2.2.6

Write $4 x^{3} - 6 x^{2} + 1$ as a set of factors.

$(2 x - 1) (2 x^{2} - 2 x - 1) = 0$

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 $0$ .

$2 x - 1 = 0$

$2 x^{2} - 2 x - 1 = 0$

Step 1.2.4

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 1.2.4.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 1.2.4.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 1.2.5

Set $2 x^{2} - 2 x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 x^{2} - 2 x - 1$ equal to $0$ .

$2 x^{2} - 2 x - 1 = 0$

Step 1.2.5.2

Solve $2 x^{2} - 2 x - 1 = 0$ 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 $a = 2$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

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

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

Step 1.2.5.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 1.2.5.2.3.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 2}$

Step 1.2.5.2.3.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.5.2.3.1.4.1

Factor $4$ out of $12$ .

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

Step 1.2.5.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.5.2.3.1.5

Pull terms out from under the radical.

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

Step 1.2.5.2.3.2

Multiply $2$ by $2$ .

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

Step 1.2.5.2.3.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{1 \pm \sqrt{3}}{2}$

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

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

Step 1.2.5.2.4.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.4.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 1.2.5.2.4.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 2}$

Step 1.2.5.2.4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.5.2.4.1.4.1

Factor $4$ out of $12$ .

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

Step 1.2.5.2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.5.2.4.1.5

Pull terms out from under the radical.

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

Step 1.2.5.2.4.2

Multiply $2$ by $2$ .

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

Step 1.2.5.2.4.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{1 \pm \sqrt{3}}{2}$

Step 1.2.5.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{3}}{2}$

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

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

Step 1.2.5.2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 1.2.5.2.5.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 2}$

Step 1.2.5.2.5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.5.2.5.1.4.1

Factor $4$ out of $12$ .

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

Step 1.2.5.2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.5.2.5.1.5

Pull terms out from under the radical.

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

Step 1.2.5.2.5.2

Multiply $2$ by $2$ .

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

Step 1.2.5.2.5.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{1 \pm \sqrt{3}}{2}$

Step 1.2.5.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{3}}{2}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 1.2.6

The final solution is all the values that make $(2 x - 1) (2 x^{2} - 2 x - 1) = 0$ true.

$x = \frac{1}{2}, \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

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 $x^{4} - 2 x^{3} + x + 1$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{1}{2}$ .

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Step 1.4.1.1

Substitute $\frac{1}{2}$ for $x$ .

${(\frac{1}{2})}^{4} - 2 {(\frac{1}{2})}^{3} + \frac{1}{2} + 1$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Remove parentheses.

${(\frac{1}{2})}^{4} - 2 {(\frac{1}{2})}^{3} + \frac{1}{2} + 1$

Step 1.4.1.2.2

Simplify each term.

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Step 1.4.1.2.2.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{4}}{2^{4}} - 2 {(\frac{1}{2})}^{3} + \frac{1}{2} + 1$

Step 1.4.1.2.2.2

One to any power is one.

$\frac{1}{2^{4}} - 2 {(\frac{1}{2})}^{3} + \frac{1}{2} + 1$

Step 1.4.1.2.2.3

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

$\frac{1}{16} - 2 {(\frac{1}{2})}^{3} + \frac{1}{2} + 1$

Step 1.4.1.2.2.4

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{16} - 2 \frac{1^{3}}{2^{3}} + \frac{1}{2} + 1$

Step 1.4.1.2.2.5

One to any power is one.

$\frac{1}{16} - 2 \frac{1}{2^{3}} + \frac{1}{2} + 1$

Step 1.4.1.2.2.6

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

$\frac{1}{16} - 2 (\frac{1}{8}) + \frac{1}{2} + 1$

Step 1.4.1.2.2.7

Cancel the common factor of $2$ .

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Step 1.4.1.2.2.7.1

Factor $2$ out of $- 2$ .

$\frac{1}{16} + 2 (- 1) \frac{1}{8} + \frac{1}{2} + 1$

Step 1.4.1.2.2.7.2

Factor $2$ out of $8$ .

$\frac{1}{16} + 2 \cdot - 1 \frac{1}{2 \cdot 4} + \frac{1}{2} + 1$

Step 1.4.1.2.2.7.3

Cancel the common factor.

$\frac{1}{16} + 2 \cdot - 1 \frac{1}{2 \cdot 4} + \frac{1}{2} + 1$

Step 1.4.1.2.2.7.4

Rewrite the expression.

$\frac{1}{16} - 1 (\frac{1}{4}) + \frac{1}{2} + 1$

Step 1.4.1.2.2.8

Rewrite $- 1 (\frac{1}{4})$ as $- (\frac{1}{4})$ .

$\frac{1}{16} - \frac{1}{4} + \frac{1}{2} + 1$

Step 1.4.1.2.3

Find the common denominator.

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Step 1.4.1.2.3.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\frac{1}{16} - (\frac{1}{4} \cdot \frac{4}{4}) + \frac{1}{2} + 1$

Step 1.4.1.2.3.2

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\frac{1}{16} - \frac{4}{4 \cdot 4} + \frac{1}{2} + 1$

Step 1.4.1.2.3.3

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$\frac{1}{16} - \frac{4}{4 \cdot 4} + \frac{1}{2} \cdot \frac{8}{8} + 1$

Step 1.4.1.2.3.4

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$\frac{1}{16} - \frac{4}{4 \cdot 4} + \frac{8}{2 \cdot 8} + 1$

Step 1.4.1.2.3.5

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

$\frac{1}{16} - \frac{4}{4 \cdot 4} + \frac{8}{2 \cdot 8} + \frac{1}{1}$

Step 1.4.1.2.3.6

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$\frac{1}{16} - \frac{4}{4 \cdot 4} + \frac{8}{2 \cdot 8} + \frac{1}{1} \cdot \frac{16}{16}$

Step 1.4.1.2.3.7

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$\frac{1}{16} - \frac{4}{4 \cdot 4} + \frac{8}{2 \cdot 8} + \frac{16}{16}$

Step 1.4.1.2.3.8

Multiply $4$ by $4$ .

$\frac{1}{16} - \frac{4}{16} + \frac{8}{2 \cdot 8} + \frac{16}{16}$

Step 1.4.1.2.3.9

Multiply $2$ by $8$ .

$\frac{1}{16} - \frac{4}{16} + \frac{8}{16} + \frac{16}{16}$

Step 1.4.1.2.4

Combine the numerators over the common denominator.

$\frac{1 - 4 + 8 + 16}{16}$

Step 1.4.1.2.5

Simplify by adding and subtracting.

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Step 1.4.1.2.5.1

Subtract $4$ from $1$ .

$\frac{- 3 + 8 + 16}{16}$

Step 1.4.1.2.5.2

Add $- 3$ and $8$ .

$\frac{5 + 16}{16}$

Step 1.4.1.2.5.3

Add $5$ and $16$ .

$\frac{21}{16}$

Step 1.4.2

Evaluate at $x = \frac{1 + \sqrt{3}}{2}$ .

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Step 1.4.2.1

Substitute $\frac{1 + \sqrt{3}}{2}$ for $x$ .

${(\frac{1 + \sqrt{3}}{2})}^{4} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Remove parentheses.

${(\frac{1 + \sqrt{3}}{2})}^{4} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2.2

Find the common denominator.

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Step 1.4.2.2.2.1

Write ${(\frac{1 + \sqrt{3}}{2})}^{4}$ as a fraction with denominator $1$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4}}{1} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2.2.2

Multiply $\frac{{(\frac{1 + \sqrt{3}}{2})}^{4}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4}}{1} \cdot \frac{2}{2} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2.2.3

Multiply $\frac{{(\frac{1 + \sqrt{3}}{2})}^{4}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2}{2} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2.2.4

Write $- 2 {(\frac{1 + \sqrt{3}}{2})}^{3}$ as a fraction with denominator $1$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3}}{1} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2.2.5

Multiply $\frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3}}{1} \cdot \frac{2}{2} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2.2.6

Multiply $\frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 + \sqrt{3}}{2} + 1$

Step 1.4.2.2.2.7

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

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 + \sqrt{3}}{2} + \frac{1}{1}$

Step 1.4.2.2.2.8

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 + \sqrt{3}}{2} + \frac{1}{1} \cdot \frac{2}{2}$

Step 1.4.2.2.2.9

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 + \sqrt{3}}{2} + \frac{2}{2}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$\frac{{(\frac{1 + \sqrt{3}}{2})}^{4} \cdot 2 - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4

Simplify each term.

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Step 1.4.2.2.4.1

Apply the product rule to $\frac{1 + \sqrt{3}}{2}$ .

$\frac{\frac{{(1 + \sqrt{3})}^{4}}{2^{4}} \cdot 2 - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.2

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

$\frac{\frac{{(1 + \sqrt{3})}^{4}}{16} \cdot 2 - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.3

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.3.1

Factor $2$ out of $16$ .

$\frac{\frac{{(1 + \sqrt{3})}^{4}}{2 (8)} \cdot 2 - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.3.2

Cancel the common factor.

$\frac{\frac{{(1 + \sqrt{3})}^{4}}{2 \cdot 8} \cdot 2 - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.3.3

Rewrite the expression.

$\frac{\frac{{(1 + \sqrt{3})}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.4

Use the Binomial Theorem.

$\frac{\frac{1^{4} + 4 \cdot 1^{3} \sqrt{3} + 6 \cdot 1^{2} {\sqrt{3}}^{2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5

Simplify each term.

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Step 1.4.2.2.4.5.1

One to any power is one.

$\frac{\frac{1 + 4 \cdot 1^{3} \sqrt{3} + 6 \cdot 1^{2} {\sqrt{3}}^{2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.2

One to any power is one.

$\frac{\frac{1 + 4 \cdot 1 \sqrt{3} + 6 \cdot 1^{2} {\sqrt{3}}^{2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.3

Multiply $4$ by $1$ .

$\frac{\frac{1 + 4 \sqrt{3} + 6 \cdot 1^{2} {\sqrt{3}}^{2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.4

One to any power is one.

$\frac{\frac{1 + 4 \sqrt{3} + 6 \cdot 1 {\sqrt{3}}^{2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.5

Multiply $6$ by $1$ .

$\frac{\frac{1 + 4 \sqrt{3} + 6 {\sqrt{3}}^{2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.2.2.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{1 + 4 \sqrt{3} + 6 {(3^{\frac{1}{2}})}^{2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.6.2

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

$\frac{\frac{1 + 4 \sqrt{3} + 6 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.6.3

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

$\frac{\frac{1 + 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.6.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.5.6.4.1

Cancel the common factor.

$\frac{\frac{1 + 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.6.4.2

Rewrite the expression.

$\frac{\frac{1 + 4 \sqrt{3} + 6 \cdot 3^{1} + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.6.5

Evaluate the exponent.

$\frac{\frac{1 + 4 \sqrt{3} + 6 \cdot 3 + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.7

Multiply $6$ by $3$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 4 \cdot 1 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.8

Multiply $4$ by $1$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 4 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.9

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

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{3^{3}} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.10

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

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{27} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.11

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.2.2.4.5.11.1

Factor $9$ out of $27$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{9 (3)} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.11.2

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{3^{2} \cdot 3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.12

Pull terms out from under the radical.

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 4 (3 \sqrt{3}) + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.13

Multiply $3$ by $4$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 1.4.2.2.4.5.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + {(3^{\frac{1}{2}})}^{4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14.2

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

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{1}{2} \cdot 4}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14.3

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

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{4}{2}}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14.4

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

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Step 1.4.2.2.4.5.14.4.1

Factor $2$ out of $4$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14.4.2

Cancel the common factors.

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Step 1.4.2.2.4.5.14.4.2.1

Factor $2$ out of $2$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14.4.2.2

Cancel the common factor.

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14.4.2.3

Rewrite the expression.

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2}{1}}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.14.4.2.4

Divide $2$ by $1$ .

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{2}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.5.15

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

$\frac{\frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 9}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.6

Add $1$ and $18$ .

$\frac{\frac{19 + 4 \sqrt{3} + 12 \sqrt{3} + 9}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.7

Add $19$ and $9$ .

$\frac{\frac{28 + 4 \sqrt{3} + 12 \sqrt{3}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.8

Add $4 \sqrt{3}$ and $12 \sqrt{3}$ .

$\frac{\frac{28 + 16 \sqrt{3}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.9

Cancel the common factor of $28 + 16 \sqrt{3}$ and $8$ .

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Step 1.4.2.2.4.9.1

Factor $4$ out of $28$ .

$\frac{\frac{4 (7) + 16 \sqrt{3}}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.9.2

Factor $4$ out of $16 \sqrt{3}$ .

$\frac{\frac{4 (7) + 4 (4 \sqrt{3})}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.9.3

Factor $4$ out of $4 (7) + 4 (4 \sqrt{3})$ .

$\frac{\frac{4 (7 + 4 \sqrt{3})}{8} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.9.4

Cancel the common factors.

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Step 1.4.2.2.4.9.4.1

Factor $4$ out of $8$ .

$\frac{\frac{4 (7 + 4 \sqrt{3})}{4 \cdot 2} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.9.4.2

Cancel the common factor.

$\frac{\frac{4 (7 + 4 \sqrt{3})}{4 \cdot 2} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.9.4.3

Rewrite the expression.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 2 {(\frac{1 + \sqrt{3}}{2})}^{3} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.10

Apply the product rule to $\frac{1 + \sqrt{3}}{2}$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 2 \frac{{(1 + \sqrt{3})}^{3}}{2^{3}} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.11

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

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 2 \frac{{(1 + \sqrt{3})}^{3}}{8} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.12

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.12.1

Factor $2$ out of $- 2$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} + 2 (- 1) \frac{{(1 + \sqrt{3})}^{3}}{8} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.12.2

Factor $2$ out of $8$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} + 2 \cdot - 1 \frac{{(1 + \sqrt{3})}^{3}}{2 \cdot 4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.12.3

Cancel the common factor.

$\frac{\frac{7 + 4 \sqrt{3}}{2} + 2 \cdot - 1 \frac{{(1 + \sqrt{3})}^{3}}{2 \cdot 4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.12.4

Rewrite the expression.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{{(1 + \sqrt{3})}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.13

Use the Binomial Theorem.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1^{3} + 3 \cdot 1^{2} \sqrt{3} + 3 \cdot 1 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14

Simplify each term.

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Step 1.4.2.2.4.14.1

One to any power is one.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \cdot 1^{2} \sqrt{3} + 3 \cdot 1 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.2

One to any power is one.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \cdot 1 \sqrt{3} + 3 \cdot 1 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.3

Multiply $3$ by $1$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 \cdot 1 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.4

Multiply $3$ by $1$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.2.2.4.14.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 {(3^{\frac{1}{2}})}^{2} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.5.2

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

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 \cdot 3^{\frac{1}{2} \cdot 2} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.5.3

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

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.5.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.14.5.4.1

Cancel the common factor.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.5.4.2

Rewrite the expression.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 \cdot 3^{1} + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.5.5

Evaluate the exponent.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 3 \cdot 3 + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.6

Multiply $3$ by $3$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 9 + {\sqrt{3}}^{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.7

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

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 9 + \sqrt{3^{3}}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.8

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

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 9 + \sqrt{27}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.9

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.2.2.4.14.9.1

Factor $9$ out of $27$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 9 + \sqrt{9 (3)}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.9.2

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 9 + \sqrt{3^{2} \cdot 3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.14.10

Pull terms out from under the radical.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \sqrt{3} + 9 + 3 \sqrt{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.15

Add $1$ and $9$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{10 + 3 \sqrt{3} + 3 \sqrt{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.16

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

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{10 + 6 \sqrt{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.17

Cancel the common factor of $10 + 6 \sqrt{3}$ and $4$ .

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Step 1.4.2.2.4.17.1

Factor $2$ out of $10$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{2 (5) + 6 \sqrt{3}}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.17.2

Factor $2$ out of $6 \sqrt{3}$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{2 (5) + 2 (3 \sqrt{3})}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.17.3

Factor $2$ out of $2 (5) + 2 (3 \sqrt{3})$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{2 (5 + 3 \sqrt{3})}{4} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.17.4

Cancel the common factors.

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Step 1.4.2.2.4.17.4.1

Factor $2$ out of $4$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{2 (5 + 3 \sqrt{3})}{2 \cdot 2} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.17.4.2

Cancel the common factor.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{2 (5 + 3 \sqrt{3})}{2 \cdot 2} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.17.4.3

Rewrite the expression.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \frac{5 + 3 \sqrt{3}}{2} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.18

Rewrite $- 1 \frac{5 + 3 \sqrt{3}}{2}$ as $- \frac{5 + 3 \sqrt{3}}{2}$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - \frac{5 + 3 \sqrt{3}}{2} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.19

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.19.1

Move the leading negative in $- \frac{5 + 3 \sqrt{3}}{2}$ into the numerator.

$\frac{\frac{7 + 4 \sqrt{3}}{2} + \frac{- (5 + 3 \sqrt{3})}{2} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.19.2

Cancel the common factor.

$\frac{\frac{7 + 4 \sqrt{3}}{2} + \frac{- (5 + 3 \sqrt{3})}{2} \cdot 2 + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.19.3

Rewrite the expression.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - (5 + 3 \sqrt{3}) + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.20

Apply the distributive property.

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 1 \cdot 5 - (3 \sqrt{3}) + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.21

Multiply $- 1$ by $5$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 5 - (3 \sqrt{3}) + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.4.22

Multiply $3$ by $- 1$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 5 - 3 \sqrt{3} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.5

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} - 5 \cdot \frac{2}{2} - 3 \sqrt{3} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.6

Combine $- 5$ and $\frac{2}{2}$ .

$\frac{\frac{7 + 4 \sqrt{3}}{2} + \frac{- 5 \cdot 2}{2} - 3 \sqrt{3} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.7

Simplify the expression.

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Step 1.4.2.2.7.1

Combine the numerators over the common denominator.

$\frac{\frac{7 + 4 \sqrt{3} - 5 \cdot 2}{2} - 3 \sqrt{3} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.7.2

Multiply $- 5$ by $2$ .

$\frac{\frac{7 + 4 \sqrt{3} - 10}{2} - 3 \sqrt{3} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.7.3

Subtract $10$ from $7$ .

$\frac{\frac{- 3 + 4 \sqrt{3}}{2} - 3 \sqrt{3} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.8

To write $- 3 \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{- 3 + 4 \sqrt{3}}{2} - 3 \sqrt{3} \cdot \frac{2}{2} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.9

Combine fractions.

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Step 1.4.2.2.9.1

Combine $- 3 \sqrt{3}$ and $\frac{2}{2}$ .

$\frac{\frac{- 3 + 4 \sqrt{3}}{2} + \frac{- 3 \sqrt{3} \cdot 2}{2} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.9.2

Combine the numerators over the common denominator.

$\frac{\frac{- 3 + 4 \sqrt{3} - 3 \sqrt{3} \cdot 2}{2} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.10

Simplify the numerator.

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Step 1.4.2.2.10.1

Multiply $2$ by $- 3$ .

$\frac{\frac{- 3 + 4 \sqrt{3} - 6 \sqrt{3}}{2} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.10.2

Subtract $6 \sqrt{3}$ from $4 \sqrt{3}$ .

$\frac{\frac{- 3 - 2 \sqrt{3}}{2} + 1 + \sqrt{3} + 2}{2}$

Step 1.4.2.2.11

Simplify the expression.

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Step 1.4.2.2.11.1

Write $1$ as a fraction with a common denominator.

$\frac{\frac{- 3 - 2 \sqrt{3}}{2} + \frac{2}{2} + \sqrt{3} + 2}{2}$

Step 1.4.2.2.11.2

Combine the numerators over the common denominator.

$\frac{\frac{- 3 - 2 \sqrt{3} + 2}{2} + \sqrt{3} + 2}{2}$

Step 1.4.2.2.11.3

Add $- 3$ and $2$ .

$\frac{\frac{- 1 - 2 \sqrt{3}}{2} + \sqrt{3} + 2}{2}$

Step 1.4.2.2.12

To write $\sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{- 1 - 2 \sqrt{3}}{2} + \sqrt{3} \cdot \frac{2}{2} + 2}{2}$

Step 1.4.2.2.13

Combine $\sqrt{3}$ and $\frac{2}{2}$ .

$\frac{\frac{- 1 - 2 \sqrt{3}}{2} + \frac{\sqrt{3} \cdot 2}{2} + 2}{2}$

Step 1.4.2.2.14

Simplify the expression.

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Step 1.4.2.2.14.1

Combine the numerators over the common denominator.

$\frac{\frac{- 1 - 2 \sqrt{3} + \sqrt{3} \cdot 2}{2} + 2}{2}$

Step 1.4.2.2.14.2

Reorder the factors of $\sqrt{3} \cdot 2$ .

$\frac{\frac{- 1 - 2 \sqrt{3} + 2 \sqrt{3}}{2} + 2}{2}$

Step 1.4.2.2.15

Add $- 2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{\frac{- 1 + 0}{2} + 2}{2}$

Step 1.4.2.2.16

Simplify the expression.

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Step 1.4.2.2.16.1

Add $- 1$ and $0$ .

$\frac{\frac{- 1}{2} + 2}{2}$

Step 1.4.2.2.16.2

Move the negative in front of the fraction.

$\frac{- \frac{1}{2} + 2}{2}$

Step 1.4.2.2.17

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- \frac{1}{2} + 2 \cdot \frac{2}{2}}{2}$

Step 1.4.2.2.18

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

$\frac{- \frac{1}{2} + \frac{2 \cdot 2}{2}}{2}$

Step 1.4.2.2.19

Combine the numerators over the common denominator.

$\frac{\frac{- 1 + 2 \cdot 2}{2}}{2}$

Step 1.4.2.2.20

Simplify the numerator.

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Step 1.4.2.2.20.1

Multiply $2$ by $2$ .

$\frac{\frac{- 1 + 4}{2}}{2}$

Step 1.4.2.2.20.2

Add $- 1$ and $4$ .

$\frac{\frac{3}{2}}{2}$

Step 1.4.2.2.21

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{2} \cdot \frac{1}{2}$

Step 1.4.2.2.22

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 1.4.2.2.22.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$\frac{3}{2 \cdot 2}$

Step 1.4.2.2.22.2

Multiply $2$ by $2$ .

$\frac{3}{4}$

Step 1.4.3

Evaluate at $x = \frac{1 - \sqrt{3}}{2}$ .

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Step 1.4.3.1

Substitute $\frac{1 - \sqrt{3}}{2}$ for $x$ .

${(\frac{1 - \sqrt{3}}{2})}^{4} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Remove parentheses.

${(\frac{1 - \sqrt{3}}{2})}^{4} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2.2

Find the common denominator.

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Step 1.4.3.2.2.1

Write ${(\frac{1 - \sqrt{3}}{2})}^{4}$ as a fraction with denominator $1$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4}}{1} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2.2.2

Multiply $\frac{{(\frac{1 - \sqrt{3}}{2})}^{4}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4}}{1} \cdot \frac{2}{2} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2.2.3

Multiply $\frac{{(\frac{1 - \sqrt{3}}{2})}^{4}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2}{2} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2.2.4

Write $- 2 {(\frac{1 - \sqrt{3}}{2})}^{3}$ as a fraction with denominator $1$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3}}{1} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2.2.5

Multiply $\frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3}}{1} \cdot \frac{2}{2} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2.2.6

Multiply $\frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3}}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 - \sqrt{3}}{2} + 1$

Step 1.4.3.2.2.7

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

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 - \sqrt{3}}{2} + \frac{1}{1}$

Step 1.4.3.2.2.8

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 - \sqrt{3}}{2} + \frac{1}{1} \cdot \frac{2}{2}$

Step 1.4.3.2.2.9

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2}{2} + \frac{- 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2}{2} + \frac{1 - \sqrt{3}}{2} + \frac{2}{2}$

Step 1.4.3.2.3

Combine the numerators over the common denominator.

$\frac{{(\frac{1 - \sqrt{3}}{2})}^{4} \cdot 2 - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4

Simplify each term.

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Step 1.4.3.2.4.1

Apply the product rule to $\frac{1 - \sqrt{3}}{2}$ .

$\frac{\frac{{(1 - \sqrt{3})}^{4}}{2^{4}} \cdot 2 - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.2

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

$\frac{\frac{{(1 - \sqrt{3})}^{4}}{16} \cdot 2 - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.3

Cancel the common factor of $2$ .

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Step 1.4.3.2.4.3.1

Factor $2$ out of $16$ .

$\frac{\frac{{(1 - \sqrt{3})}^{4}}{2 (8)} \cdot 2 - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.3.2

Cancel the common factor.

$\frac{\frac{{(1 - \sqrt{3})}^{4}}{2 \cdot 8} \cdot 2 - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.3.3

Rewrite the expression.

$\frac{\frac{{(1 - \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.4

Use the Binomial Theorem.

$\frac{\frac{1^{4} + 4 \cdot 1^{3} (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5

Simplify each term.

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Step 1.4.3.2.4.5.1

One to any power is one.

$\frac{\frac{1 + 4 \cdot 1^{3} (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.2

One to any power is one.

$\frac{\frac{1 + 4 \cdot 1 (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.3

Multiply $4$ by $1$ .

$\frac{\frac{1 + 4 (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.4

Multiply $- 1$ by $4$ .

$\frac{\frac{1 - 4 \sqrt{3} + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.5

One to any power is one.

$\frac{\frac{1 - 4 \sqrt{3} + 6 \cdot 1 {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.6

Multiply $6$ by $1$ .

$\frac{\frac{1 - 4 \sqrt{3} + 6 {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.7

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

$\frac{\frac{1 - 4 \sqrt{3} + 6 ({(- 1)}^{2} {\sqrt{3}}^{2}) + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.8

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

$\frac{\frac{1 - 4 \sqrt{3} + 6 (1 {\sqrt{3}}^{2}) + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.9

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$\frac{\frac{1 - 4 \sqrt{3} + 6 {\sqrt{3}}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.10

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.3.2.4.5.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{1 - 4 \sqrt{3} + 6 {(3^{\frac{1}{2}})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.10.2

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

$\frac{\frac{1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.10.3

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

$\frac{\frac{1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.10.4

Cancel the common factor of $2$ .

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Step 1.4.3.2.4.5.10.4.1

Cancel the common factor.

$\frac{\frac{1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.10.4.2

Rewrite the expression.

$\frac{\frac{1 - 4 \sqrt{3} + 6 \cdot 3^{1} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.10.5

Evaluate the exponent.

$\frac{\frac{1 - 4 \sqrt{3} + 6 \cdot 3 + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.11

Multiply $6$ by $3$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.12

Multiply $4$ by $1$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.13

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 ({(- 1)}^{3} {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.14

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 (- {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.15

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{3^{3}}) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.16

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{27}) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.17

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.3.2.4.5.17.1

Factor $9$ out of $27$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{9 (3)}) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.17.2

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{3^{2} \cdot 3}) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.18

Pull terms out from under the radical.

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 (- (3 \sqrt{3})) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.19

Multiply $3$ by $- 1$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 + 4 (- 3 \sqrt{3}) + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.20

Multiply $- 3$ by $4$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(- \sqrt{3})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.21

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(- 1)}^{4} {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.22

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 1 {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.23

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {\sqrt{3}}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 1.4.3.2.4.5.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(3^{\frac{1}{2}})}^{4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24.2

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{1}{2} \cdot 4}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24.3

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{4}{2}}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24.4

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

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Step 1.4.3.2.4.5.24.4.1

Factor $2$ out of $4$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24.4.2

Cancel the common factors.

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Step 1.4.3.2.4.5.24.4.2.1

Factor $2$ out of $2$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24.4.2.2

Cancel the common factor.

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24.4.2.3

Rewrite the expression.

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2}{1}}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.24.4.2.4

Divide $2$ by $1$ .

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{2}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.5.25

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

$\frac{\frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 9}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.6

Add $1$ and $18$ .

$\frac{\frac{19 - 4 \sqrt{3} - 12 \sqrt{3} + 9}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.7

Add $19$ and $9$ .

$\frac{\frac{28 - 4 \sqrt{3} - 12 \sqrt{3}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.8

Subtract $12 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$\frac{\frac{28 - 16 \sqrt{3}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.9

Cancel the common factor of $28 - 16 \sqrt{3}$ and $8$ .

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Step 1.4.3.2.4.9.1

Factor $4$ out of $28$ .

$\frac{\frac{4 (7) - 16 \sqrt{3}}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.9.2

Factor $4$ out of $- 16 \sqrt{3}$ .

$\frac{\frac{4 (7) + 4 (- 4 \sqrt{3})}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.9.3

Factor $4$ out of $4 (7) + 4 (- 4 \sqrt{3})$ .

$\frac{\frac{4 (7 - 4 \sqrt{3})}{8} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.9.4

Cancel the common factors.

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Step 1.4.3.2.4.9.4.1

Factor $4$ out of $8$ .

$\frac{\frac{4 (7 - 4 \sqrt{3})}{4 \cdot 2} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.9.4.2

Cancel the common factor.

$\frac{\frac{4 (7 - 4 \sqrt{3})}{4 \cdot 2} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.9.4.3

Rewrite the expression.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 2 {(\frac{1 - \sqrt{3}}{2})}^{3} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.10

Apply the product rule to $\frac{1 - \sqrt{3}}{2}$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 2 \frac{{(1 - \sqrt{3})}^{3}}{2^{3}} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.11

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 2 \frac{{(1 - \sqrt{3})}^{3}}{8} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.12

Cancel the common factor of $2$ .

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Step 1.4.3.2.4.12.1

Factor $2$ out of $- 2$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} + 2 (- 1) \frac{{(1 - \sqrt{3})}^{3}}{8} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.12.2

Factor $2$ out of $8$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} + 2 \cdot - 1 \frac{{(1 - \sqrt{3})}^{3}}{2 \cdot 4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.12.3

Cancel the common factor.

$\frac{\frac{7 - 4 \sqrt{3}}{2} + 2 \cdot - 1 \frac{{(1 - \sqrt{3})}^{3}}{2 \cdot 4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.12.4

Rewrite the expression.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{{(1 - \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.13

Use the Binomial Theorem.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1^{3} + 3 \cdot 1^{2} (- \sqrt{3}) + 3 \cdot 1 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14

Simplify each term.

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Step 1.4.3.2.4.14.1

One to any power is one.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \cdot 1^{2} (- \sqrt{3}) + 3 \cdot 1 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.2

One to any power is one.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 + 3 \cdot 1 (- \sqrt{3}) + 3 \cdot 1 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.3

Multiply $3$ by $1$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 + 3 (- \sqrt{3}) + 3 \cdot 1 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.4

Multiply $- 1$ by $3$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 \cdot 1 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.5

Multiply $3$ by $1$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.6

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 ({(- 1)}^{2} {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.7

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 (1 {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.8

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 {\sqrt{3}}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.9

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.3.2.4.14.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 {(3^{\frac{1}{2}})}^{2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.9.2

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 \cdot 3^{\frac{1}{2} \cdot 2} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.9.3

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.9.4

Cancel the common factor of $2$ .

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Step 1.4.3.2.4.14.9.4.1

Cancel the common factor.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.9.4.2

Rewrite the expression.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 \cdot 3^{1} + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.9.5

Evaluate the exponent.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 3 \cdot 3 + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.10

Multiply $3$ by $3$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 + {(- \sqrt{3})}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.11

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 + {(- 1)}^{3} {\sqrt{3}}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.12

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 - {\sqrt{3}}^{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.13

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 - \sqrt{3^{3}}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.14

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 - \sqrt{27}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.15

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.3.2.4.14.15.1

Factor $9$ out of $27$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 - \sqrt{9 (3)}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.15.2

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 - \sqrt{3^{2} \cdot 3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.16

Pull terms out from under the radical.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 - (3 \sqrt{3})}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.14.17

Multiply $3$ by $- 1$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{1 - 3 \sqrt{3} + 9 - 3 \sqrt{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.15

Add $1$ and $9$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{10 - 3 \sqrt{3} - 3 \sqrt{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.16

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

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{10 - 6 \sqrt{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.17

Cancel the common factor of $10 - 6 \sqrt{3}$ and $4$ .

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Step 1.4.3.2.4.17.1

Factor $2$ out of $10$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{2 (5) - 6 \sqrt{3}}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.17.2

Factor $2$ out of $- 6 \sqrt{3}$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{2 (5) + 2 (- 3 \sqrt{3})}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.17.3

Factor $2$ out of $2 (5) + 2 (- 3 \sqrt{3})$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{2 (5 - 3 \sqrt{3})}{4} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.17.4

Cancel the common factors.

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Step 1.4.3.2.4.17.4.1

Factor $2$ out of $4$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{2 (5 - 3 \sqrt{3})}{2 \cdot 2} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.17.4.2

Cancel the common factor.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{2 (5 - 3 \sqrt{3})}{2 \cdot 2} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.17.4.3

Rewrite the expression.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \frac{5 - 3 \sqrt{3}}{2} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.18

Rewrite $- 1 \frac{5 - 3 \sqrt{3}}{2}$ as $- \frac{5 - 3 \sqrt{3}}{2}$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - \frac{5 - 3 \sqrt{3}}{2} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.19

Cancel the common factor of $2$ .

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Step 1.4.3.2.4.19.1

Move the leading negative in $- \frac{5 - 3 \sqrt{3}}{2}$ into the numerator.

$\frac{\frac{7 - 4 \sqrt{3}}{2} + \frac{- (5 - 3 \sqrt{3})}{2} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.19.2

Cancel the common factor.

$\frac{\frac{7 - 4 \sqrt{3}}{2} + \frac{- (5 - 3 \sqrt{3})}{2} \cdot 2 + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.19.3

Rewrite the expression.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - (5 - 3 \sqrt{3}) + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.20

Apply the distributive property.

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 1 \cdot 5 - (- 3 \sqrt{3}) + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.21

Multiply $- 1$ by $5$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 5 - (- 3 \sqrt{3}) + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.4.22

Multiply $- 3$ by $- 1$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 5 + 3 \sqrt{3} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.5

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} - 5 \cdot \frac{2}{2} + 3 \sqrt{3} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.6

Combine $- 5$ and $\frac{2}{2}$ .

$\frac{\frac{7 - 4 \sqrt{3}}{2} + \frac{- 5 \cdot 2}{2} + 3 \sqrt{3} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.7

Simplify the expression.

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Step 1.4.3.2.7.1

Combine the numerators over the common denominator.

$\frac{\frac{7 - 4 \sqrt{3} - 5 \cdot 2}{2} + 3 \sqrt{3} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.7.2

Multiply $- 5$ by $2$ .

$\frac{\frac{7 - 4 \sqrt{3} - 10}{2} + 3 \sqrt{3} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.7.3

Subtract $10$ from $7$ .

$\frac{\frac{- 3 - 4 \sqrt{3}}{2} + 3 \sqrt{3} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.8

To write $3 \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{- 3 - 4 \sqrt{3}}{2} + 3 \sqrt{3} \cdot \frac{2}{2} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.9

Combine fractions.

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Step 1.4.3.2.9.1

Combine $3 \sqrt{3}$ and $\frac{2}{2}$ .

$\frac{\frac{- 3 - 4 \sqrt{3}}{2} + \frac{3 \sqrt{3} \cdot 2}{2} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.9.2

Combine the numerators over the common denominator.

$\frac{\frac{- 3 - 4 \sqrt{3} + 3 \sqrt{3} \cdot 2}{2} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.10

Simplify the numerator.

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Step 1.4.3.2.10.1

Multiply $2$ by $3$ .

$\frac{\frac{- 3 - 4 \sqrt{3} + 6 \sqrt{3}}{2} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.10.2

Add $- 4 \sqrt{3}$ and $6 \sqrt{3}$ .

$\frac{\frac{- 3 + 2 \sqrt{3}}{2} + 1 - \sqrt{3} + 2}{2}$

Step 1.4.3.2.11

Simplify the expression.

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Step 1.4.3.2.11.1

Write $1$ as a fraction with a common denominator.

$\frac{\frac{- 3 + 2 \sqrt{3}}{2} + \frac{2}{2} - \sqrt{3} + 2}{2}$

Step 1.4.3.2.11.2

Combine the numerators over the common denominator.

$\frac{\frac{- 3 + 2 \sqrt{3} + 2}{2} - \sqrt{3} + 2}{2}$

Step 1.4.3.2.11.3

Add $- 3$ and $2$ .

$\frac{\frac{- 1 + 2 \sqrt{3}}{2} - \sqrt{3} + 2}{2}$

Step 1.4.3.2.12

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{- 1 + 2 \sqrt{3}}{2} - \sqrt{3} \cdot \frac{2}{2} + 2}{2}$

Step 1.4.3.2.13

Combine fractions.

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Step 1.4.3.2.13.1

Combine $- \sqrt{3}$ and $\frac{2}{2}$ .

$\frac{\frac{- 1 + 2 \sqrt{3}}{2} + \frac{- \sqrt{3} \cdot 2}{2} + 2}{2}$

Step 1.4.3.2.13.2

Combine the numerators over the common denominator.

$\frac{\frac{- 1 + 2 \sqrt{3} - \sqrt{3} \cdot 2}{2} + 2}{2}$

Step 1.4.3.2.14

Simplify each term.

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Step 1.4.3.2.14.1

Simplify the numerator.

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Step 1.4.3.2.14.1.1

Multiply $2$ by $- 1$ .

$\frac{\frac{- 1 + 2 \sqrt{3} - 2 \sqrt{3}}{2} + 2}{2}$

Step 1.4.3.2.14.1.2

Subtract $2 \sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{\frac{- 1 + 0}{2} + 2}{2}$

Step 1.4.3.2.14.1.3

Add $- 1$ and $0$ .

$\frac{\frac{- 1}{2} + 2}{2}$

Step 1.4.3.2.14.2

Move the negative in front of the fraction.

$\frac{- \frac{1}{2} + 2}{2}$

Step 1.4.3.2.15

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- \frac{1}{2} + 2 \cdot \frac{2}{2}}{2}$

Step 1.4.3.2.16

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

$\frac{- \frac{1}{2} + \frac{2 \cdot 2}{2}}{2}$

Step 1.4.3.2.17

Combine the numerators over the common denominator.

$\frac{\frac{- 1 + 2 \cdot 2}{2}}{2}$

Step 1.4.3.2.18

Simplify the numerator.

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Step 1.4.3.2.18.1

Multiply $2$ by $2$ .

$\frac{\frac{- 1 + 4}{2}}{2}$

Step 1.4.3.2.18.2

Add $- 1$ and $4$ .

$\frac{\frac{3}{2}}{2}$

Step 1.4.3.2.19

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{2} \cdot \frac{1}{2}$

Step 1.4.3.2.20

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 1.4.3.2.20.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$\frac{3}{2 \cdot 2}$

Step 1.4.3.2.20.2

Multiply $2$ by $2$ .

$\frac{3}{4}$

Step 1.4.4

List all of the points.

$(\frac{1}{2}, \frac{21}{16}), (\frac{1 + \sqrt{3}}{2}, \frac{3}{4}), (\frac{1 - \sqrt{3}}{2}, \frac{3}{4})$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 1$ .

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Step 2.1.1

Substitute $- 1$ for $x$ .

${(- 1)}^{4} - 2 {(- 1)}^{3} - 1 + 1$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Remove parentheses.

${(- 1)}^{4} - 2 {(- 1)}^{3} - 1 + 1$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

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

$1 - 2 {(- 1)}^{3} - 1 + 1$

Step 2.1.2.2.2

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

$1 - 2 \cdot - 1 - 1 + 1$

Step 2.1.2.2.3

Multiply $- 2$ by $- 1$ .

$1 + 2 - 1 + 1$

Step 2.1.2.3

Simplify by adding and subtracting.

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Step 2.1.2.3.1

Add $1$ and $2$ .

$3 - 1 + 1$

Step 2.1.2.3.2

Subtract $1$ from $3$ .

$2 + 1$

Step 2.1.2.3.3

Add $2$ and $1$ .

$3$

Step 2.2

Evaluate at $x = 3$ .

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Step 2.2.1

Substitute $3$ for $x$ .

${(3)}^{4} - 2 {(3)}^{3} + 3 + 1$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Remove parentheses.

${(3)}^{4} - 2 {(3)}^{3} + 3 + 1$

Step 2.2.2.2

Simplify each term.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

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

$81 - 2 \cdot 27 + 3 + 1$

Step 2.2.2.2.3

Multiply $- 2$ by $27$ .

$81 - 54 + 3 + 1$

Step 2.2.2.3

Simplify by adding and subtracting.

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Step 2.2.2.3.1

Subtract $54$ from $81$ .

$27 + 3 + 1$

Step 2.2.2.3.2

Add $27$ and $3$ .

$30 + 1$

Step 2.2.2.3.3

Add $30$ and $1$ .

$31$

Step 2.3

List all of the points.

$(- 1, 3), (3, 31)$

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.

Absolute Maximum: $(3, 31)$

Absolute Minimum: $(\frac{1 + \sqrt{3}}{2}, \frac{3}{4}), (\frac{1 - \sqrt{3}}{2}, \frac{3}{4})$

Step 4