Calculus Examples

$f (x) = \frac{x + 2}{x^{2} + 4}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 2$ and $g (x) = x^{2} + 4$ .

$\frac{(x^{2} + 4) \frac{d}{d x} [x + 2] - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$\frac{(x^{2} + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [2]) - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 1.2.2

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

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

Step 1.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 4) (1 + 0) - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

$\frac{(x^{2} + 4) \cdot 1 - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 1.2.4.2

Multiply $x^{2} + 4$ by $1$ .

$\frac{x^{2} + 4 - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 1.2.5

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

$\frac{x^{2} + 4 - (x + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])}{{(x^{2} + 4)}^{2}}$

Step 1.2.6

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

$\frac{x^{2} + 4 - (x + 2) (2 x + \frac{d}{d x} [4])}{{(x^{2} + 4)}^{2}}$

Step 1.2.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x^{2} + 4 - (x + 2) (2 x + 0)}{{(x^{2} + 4)}^{2}}$

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $2 x$ and $0$ .

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

Step 1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{x^{2} + 4 + (- 2 x - 2 \cdot 2) x}{{(x^{2} + 4)}^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{x^{2} + 4 - 2 x \cdot x - 2 \cdot 2 x}{{(x^{2} + 4)}^{2}}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.3.1.1.1

Move $x$ .

$\frac{x^{2} + 4 - 2 (x \cdot x) - 2 \cdot 2 x}{{(x^{2} + 4)}^{2}}$

Step 1.3.3.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 4 - 2 x^{2} - 2 \cdot 2 x}{{(x^{2} + 4)}^{2}}$

Step 1.3.3.1.2

Multiply $- 2$ by $2$ .

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

Step 1.3.3.2

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 1.3.4

Reorder terms.

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

Step 1.3.5

Factor $- 1$ out of $- x^{2}$ .

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

Step 1.3.6

Factor $- 1$ out of $- 4 x$ .

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

Step 1.3.7

Factor $- 1$ out of $- (x^{2}) - (4 x)$ .

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

Step 1.3.8

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 1.3.9

Factor $- 1$ out of $- (x^{2} + 4 x) - 1 (- 4)$ .

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

Step 1.3.10

Rewrite $- (x^{2} + 4 x - 4)$ as $- 1 (x^{2} + 4 x - 4)$ .

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

Step 1.3.11

Move the negative in front of the fraction.

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

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}}$ .

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

Step 2.2

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply the exponents in ${({(x^{2} + 4)}^{2})}^{2}$ .

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $2$ by $2$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Multiply $4$ by $1$ .

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

Step 2.3.7

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{{(x^{2} + 4)}^{2} (2 x + 4 + 0) - (x^{2} + 4 x - 4) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}} + \frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.3.8

Add $2 x + 4$ and $0$ .

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

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 4$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 4$ .

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

Step 2.4.2

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

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

Step 2.4.3

Replace all occurrences of $u$ with $x^{2} + 4$ .

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

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

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

Step 2.5.2

Factor $x^{2} + 4$ out of ${(x^{2} + 4)}^{2} (2 x + 4) - 2 (x^{2} + 4 x - 4) ((x^{2} + 4) \frac{d}{d x} [x^{2} + 4])$ .

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Step 2.5.2.1

Factor $x^{2} + 4$ out of ${(x^{2} + 4)}^{2} (2 x + 4)$ .

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

Step 2.5.2.2

Factor $x^{2} + 4$ out of $- 2 (x^{2} + 4 x - 4) ((x^{2} + 4) \frac{d}{d x} [x^{2} + 4])$ .

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

Step 2.5.2.3

Factor $x^{2} + 4$ out of $(x^{2} + 4) ((x^{2} + 4) (2 x + 4)) + (x^{2} + 4) (- 2 (x^{2} + 4 x - 4) (\frac{d}{d x} [x^{2} + 4]))$ .

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

Step 2.6

Cancel the common factors.

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Step 2.6.1

Factor $x^{2} + 4$ out of ${(x^{2} + 4)}^{4}$ .

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

Step 2.6.2

Cancel the common factor.

Step 2.6.3

Rewrite the expression.

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{(x^{2} + 4) (2 x + 4) - 2 (x^{2} + 4 x - 4) (2 x + 0)}{{(x^{2} + 4)}^{3}} + \frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}} \cdot \frac{d}{d x} (- 1)$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

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

Step 2.10.2

Multiply $2$ by $- 2$ .

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

Step 2.11

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

$f'' (x) = - \frac{(x^{2} + 4) (2 x + 4) - 4 (x^{2} + 4 x - 4) x}{{(x^{2} + 4)}^{3}} + \frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}} \cdot 0$

Step 2.12

Simplify the expression.

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Step 2.12.1

Multiply $\frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}}$ by $0$ .

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

Step 2.12.2

Add $- \frac{(x^{2} + 4) (2 x + 4) - 4 (x^{2} + 4 x - 4) x}{{(x^{2} + 4)}^{3}}$ and $0$ .

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

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

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

Step 2.13.2

Apply the distributive property.

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

Step 2.13.3

Simplify the numerator.

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Step 2.13.3.1

Simplify each term.

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Step 2.13.3.1.1

Expand $(x^{2} + 4) (2 x + 4)$ using the FOIL Method.

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Step 2.13.3.1.1.1

Apply the distributive property.

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

Step 2.13.3.1.1.2

Apply the distributive property.

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

Step 2.13.3.1.1.3

Apply the distributive property.

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

Step 2.13.3.1.2

Simplify each term.

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Step 2.13.3.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.13.3.1.2.2

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

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Step 2.13.3.1.2.2.1

Move $x$ .

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

Step 2.13.3.1.2.2.2

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

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Step 2.13.3.1.2.2.2.1

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

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

Step 2.13.3.1.2.2.2.2

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

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

Step 2.13.3.1.2.2.3

Add $1$ and $2$ .

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

Step 2.13.3.1.2.3

Move $4$ to the left of $x^{2}$ .

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

Step 2.13.3.1.2.4

Multiply $2$ by $4$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 4 \cdot 4 - 4 x^{2} x - 4 (4 x) x - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.2.5

Multiply $4$ by $4$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 x^{2} x - 4 (4 x) x - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.3

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

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Step 2.13.3.1.3.1

Move $x$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 (x \cdot x^{2}) - 4 (4 x) x - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.3.2

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

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Step 2.13.3.1.3.2.1

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

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 (x \cdot x^{2}) - 4 (4 x) x - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.3.2.2

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

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 x^{1 + 2} - 4 (4 x) x - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.3.3

Add $1$ and $2$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 x^{3} - 4 (4 x) x - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.13.3.1.4.1

Move $x$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 x^{3} - 4 (4 (x \cdot x)) - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.4.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 x^{3} - 4 (4 x^{2}) - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.5

Multiply $4$ by $- 4$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 x^{3} - 16 x^{2} - 4 \cdot (- 4 x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.1.6

Multiply $- 4$ by $- 4$ .

$f'' (x) = - \frac{2 x^{3} + 4 x^{2} + 8 x + 16 - 4 x^{3} - 16 x^{2} + 16 x}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.2

Subtract $4 x^{3}$ from $2 x^{3}$ .

$f'' (x) = - \frac{- 2 x^{3} + 4 x^{2} + 8 x + 16 - 16 x^{2} + 16 x}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.3

Subtract $16 x^{2}$ from $4 x^{2}$ .

$f'' (x) = - \frac{- 2 x^{3} - 12 x^{2} + 8 x + 16 + 16 x}{{(x^{2} + 4)}^{3}}$

Step 2.13.3.4

Add $8 x$ and $16 x$ .

$f'' (x) = - \frac{- 2 x^{3} - 12 x^{2} + 24 x + 16}{{(x^{2} + 4)}^{3}}$

Step 2.13.4

Factor $2$ out of $- 2 x^{3} - 12 x^{2} + 24 x + 16$ .

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Step 2.13.4.1

Factor $2$ out of $- 2 x^{3}$ .

$f'' (x) = - \frac{2 (- x^{3}) - 12 x^{2} + 24 x + 16}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.2

Factor $2$ out of $- 12 x^{2}$ .

$f'' (x) = - \frac{2 (- x^{3}) + 2 (- 6 x^{2}) + 24 x + 16}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.3

Factor $2$ out of $24 x$ .

$f'' (x) = - \frac{2 (- x^{3}) + 2 (- 6 x^{2}) + 2 (12 x) + 16}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.4

Factor $2$ out of $16$ .

$f'' (x) = - \frac{2 (- x^{3}) + 2 (- 6 x^{2}) + 2 (12 x) + 2 (8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.5

Factor $2$ out of $2 (- x^{3}) + 2 (- 6 x^{2})$ .

$f'' (x) = - \frac{2 (- x^{3} - 6 x^{2}) + 2 (12 x) + 2 (8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.6

Factor $2$ out of $2 (- x^{3} - 6 x^{2}) + 2 (12 x)$ .

$f'' (x) = - \frac{2 (- x^{3} - 6 x^{2} + 12 x) + 2 (8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.7

Factor $2$ out of $2 (- x^{3} - 6 x^{2} + 12 x) + 2 (8)$ .

$f'' (x) = - \frac{2 (- x^{3} - 6 x^{2} + 12 x + 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.5

Factor $- 1$ out of $- x^{3}$ .

$f'' (x) = - \frac{2 (- (x^{3}) - 6 x^{2} + 12 x + 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.6

Factor $- 1$ out of $- 6 x^{2}$ .

$f'' (x) = - \frac{2 (- (x^{3}) - (6 x^{2}) + 12 x + 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.7

Factor $- 1$ out of $- (x^{3}) - (6 x^{2})$ .

$f'' (x) = - \frac{2 (- (x^{3} + 6 x^{2}) + 12 x + 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.8

Factor $- 1$ out of $12 x$ .

$f'' (x) = - \frac{2 (- (x^{3} + 6 x^{2}) - (- 12 x) + 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.9

Factor $- 1$ out of $- (x^{3} + 6 x^{2}) - (- 12 x)$ .

$f'' (x) = - \frac{2 (- (x^{3} + 6 x^{2} - 12 x) + 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.10

Rewrite $8$ as $- 1 (- 8)$ .

$f'' (x) = - \frac{2 (- (x^{3} + 6 x^{2} - 12 x) - 1 \cdot - 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.11

Factor $- 1$ out of $- (x^{3} + 6 x^{2} - 12 x) - 1 (- 8)$ .

$f'' (x) = - \frac{2 (- (x^{3} + 6 x^{2} - 12 x - 8))}{{(x^{2} + 4)}^{3}}$

Step 2.13.12

Rewrite $- (x^{3} + 6 x^{2} - 12 x - 8)$ as $- 1 (x^{3} + 6 x^{2} - 12 x - 8)$ .

$f'' (x) = - \frac{2 (- 1 (x^{3} + 6 x^{2} - 12 x - 8))}{{(x^{2} + 4)}^{3}}$

Step 2.13.13

Move the negative in front of the fraction.

$f'' (x) = \frac{2 (x^{3} + 6 x^{2} - 12 x - 8)}{{(x^{2} + 4)}^{3}}$

Step 2.13.14

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{2 (x^{3} + 6 x^{2} - 12 x - 8)}{{(x^{2} + 4)}^{3}})$

Step 2.13.15

Multiply $\frac{2 (x^{3} + 6 x^{2} - 12 x - 8)}{{(x^{2} + 4)}^{3}}$ by $1$ .

$f'' (x) = \frac{2 (x^{3} + 6 x^{2} - 12 x - 8)}{{(x^{2} + 4)}^{3}}$

Step 3

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

$- \frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{(x^{2} + 4) \frac{d}{d x} [x + 2] - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$\frac{(x^{2} + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [2]) - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.1.2.2

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

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

Step 4.1.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 4) (1 + 0) - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.1.2.4

Simplify the expression.

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Step 4.1.2.4.1

Add $1$ and $0$ .

$\frac{(x^{2} + 4) \cdot 1 - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.1.2.4.2

Multiply $x^{2} + 4$ by $1$ .

$\frac{x^{2} + 4 - (x + 2) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.1.2.5

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

$\frac{x^{2} + 4 - (x + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])}{{(x^{2} + 4)}^{2}}$

Step 4.1.2.6

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

$\frac{x^{2} + 4 - (x + 2) (2 x + \frac{d}{d x} [4])}{{(x^{2} + 4)}^{2}}$

Step 4.1.2.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x^{2} + 4 - (x + 2) (2 x + 0)}{{(x^{2} + 4)}^{2}}$

Step 4.1.2.8

Simplify the expression.

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Step 4.1.2.8.1

Add $2 x$ and $0$ .

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

Step 4.1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 4.1.3

Simplify.

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Step 4.1.3.1

Apply the distributive property.

$\frac{x^{2} + 4 + (- 2 x - 2 \cdot 2) x}{{(x^{2} + 4)}^{2}}$

Step 4.1.3.2

Apply the distributive property.

$\frac{x^{2} + 4 - 2 x \cdot x - 2 \cdot 2 x}{{(x^{2} + 4)}^{2}}$

Step 4.1.3.3

Simplify the numerator.

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Step 4.1.3.3.1

Simplify each term.

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Step 4.1.3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.3.3.1.1.1

Move $x$ .

$\frac{x^{2} + 4 - 2 (x \cdot x) - 2 \cdot 2 x}{{(x^{2} + 4)}^{2}}$

Step 4.1.3.3.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 4 - 2 x^{2} - 2 \cdot 2 x}{{(x^{2} + 4)}^{2}}$

Step 4.1.3.3.1.2

Multiply $- 2$ by $2$ .

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

Step 4.1.3.3.2

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 4.1.3.4

Reorder terms.

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

Step 4.1.3.5

Factor $- 1$ out of $- x^{2}$ .

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

Step 4.1.3.6

Factor $- 1$ out of $- 4 x$ .

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

Step 4.1.3.7

Factor $- 1$ out of $- (x^{2}) - (4 x)$ .

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

Step 4.1.3.8

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 4.1.3.9

Factor $- 1$ out of $- (x^{2} + 4 x) - 1 (- 4)$ .

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

Step 4.1.3.10

Rewrite $- (x^{2} + 4 x - 4)$ as $- 1 (x^{2} + 4 x - 4)$ .

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

Step 4.1.3.11

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{x^{2} + 4 x - 4}{{(x^{2} + 4)}^{2}} = 0$

Step 5.2

Set the numerator equal to zero.

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

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Use the quadratic formula to find the solutions.

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

Step 5.3.2

Substitute the values $a = 1$ , $b = 4$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 5.3.3

Simplify.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 5.3.3.1.2

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

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Step 5.3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 5.3.3.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 5.3.3.1.3

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Step 5.3.3.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 5.3.3.1.4.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 5.3.3.1.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 5.3.3.1.5

Pull terms out from under the radical.

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

Step 5.3.3.2

Multiply $2$ by $1$ .

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

Step 5.3.3.3

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

$x = - 2 \pm 2 \sqrt{2}$

Step 5.3.4

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

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Step 5.3.4.1

Simplify the numerator.

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Step 5.3.4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 5.3.4.1.2

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

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Step 5.3.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 5.3.4.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 5.3.4.1.3

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Step 5.3.4.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 5.3.4.1.4.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 5.3.4.1.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 5.3.4.1.5

Pull terms out from under the radical.

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

Step 5.3.4.2

Multiply $2$ by $1$ .

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

Step 5.3.4.3

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

$x = - 2 \pm 2 \sqrt{2}$

Step 5.3.4.4

Change the $\pm$ to $+$ .

$x = - 2 + 2 \sqrt{2}$

Step 5.3.5

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

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Step 5.3.5.1

Simplify the numerator.

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Step 5.3.5.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 5.3.5.1.2

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

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Step 5.3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 5.3.5.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 5.3.5.1.3

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Step 5.3.5.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 5.3.5.1.4.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 5.3.5.1.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 5.3.5.1.5

Pull terms out from under the radical.

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

Step 5.3.5.2

Multiply $2$ by $1$ .

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

Step 5.3.5.3

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

$x = - 2 \pm 2 \sqrt{2}$

Step 5.3.5.4

Change the $\pm$ to $-$ .

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

Step 5.3.6

The final answer is the combination of both solutions.

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

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

Step 7

Critical points to evaluate.

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

Step 8

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

$\frac{2 ({(- 2 + 2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Use the Binomial Theorem.

$\frac{2 ({(- 2)}^{3} + 3 {(- 2)}^{2} (2 \sqrt{2}) + 3 \cdot - 2 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2

Simplify each term.

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Step 9.1.2.1

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

$\frac{2 (- 8 + 3 {(- 2)}^{2} (2 \sqrt{2}) + 3 \cdot - 2 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.2

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

$\frac{2 (- 8 + 3 \cdot 4 (2 \sqrt{2}) + 3 \cdot - 2 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.3

Multiply $3$ by $4$ .

$\frac{2 (- 8 + 12 (2 \sqrt{2}) + 3 \cdot - 2 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.4

Multiply $2$ by $12$ .

$\frac{2 (- 8 + 24 \sqrt{2} + 3 \cdot - 2 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.5

Multiply $3$ by $- 2$ .

$\frac{2 (- 8 + 24 \sqrt{2} - 6 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.6

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

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (2^{2} {\sqrt{2}}^{2}) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.7

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

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (4 {\sqrt{2}}^{2}) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.8

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

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Step 9.1.2.8.1

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

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (4 {(2^{\frac{1}{2}})}^{2}) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.8.2

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

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (4 \cdot 2^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.8.3

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

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (4 \cdot 2^{\frac{2}{2}}) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.8.4

Cancel the common factor of $2$ .

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Step 9.1.2.8.4.1

Cancel the common factor.

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (4 \cdot 2^{\frac{2}{2}}) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.8.4.2

Rewrite the expression.

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (4 \cdot 2^{1}) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.8.5

Evaluate the exponent.

$\frac{2 (- 8 + 24 \sqrt{2} - 6 (4 \cdot 2) + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.9

Multiply $- 6 (4 \cdot 2)$ .

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Step 9.1.2.9.1

Multiply $4$ by $2$ .

$\frac{2 (- 8 + 24 \sqrt{2} - 6 \cdot 8 + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.9.2

Multiply $- 6$ by $8$ .

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + {(2 \sqrt{2})}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.10

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

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 2^{3} {\sqrt{2}}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.11

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

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 8 {\sqrt{2}}^{3} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.12

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

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 8 \sqrt{2^{3}} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.13

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

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 8 \sqrt{8} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.14

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 9.1.2.14.1

Factor $4$ out of $8$ .

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 8 \sqrt{4 (2)} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.14.2

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

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 8 \sqrt{2^{2} \cdot 2} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.15

Pull terms out from under the radical.

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 8 (2 \sqrt{2}) + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.2.16

Multiply $2$ by $8$ .

$\frac{2 (- 8 + 24 \sqrt{2} - 48 + 16 \sqrt{2} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.3

Subtract $48$ from $- 8$ .

$\frac{2 (- 56 + 24 \sqrt{2} + 16 \sqrt{2} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.4

Add $24 \sqrt{2}$ and $16 \sqrt{2}$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 {(- 2 + 2 \sqrt{2})}^{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.5

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

$\frac{2 (- 56 + 40 \sqrt{2} + 6 ((- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.6

Expand $(- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2})$ using the FOIL Method.

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Step 9.1.6.1

Apply the distributive property.

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (- 2 (- 2 + 2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.6.2

Apply the distributive property.

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.6.3

Apply the distributive property.

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7

Simplify and combine like terms.

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Step 9.1.7.1

Simplify each term.

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Step 9.1.7.1.1

Multiply $- 2$ by $- 2$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.2

Multiply $2$ by $- 2$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.3

Multiply $- 2$ by $2$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 2 \sqrt{2} (2 \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.4

Multiply $2 \sqrt{2} (2 \sqrt{2})$ .

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Step 9.1.7.1.4.1

Multiply $2$ by $2$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} \sqrt{2})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.4.4

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

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.4.5

Add $1$ and $1$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {\sqrt{2}}^{2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.5

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

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Step 9.1.7.1.5.1

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

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.5.2

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

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.5.3

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

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 9.1.7.1.5.4.1

Cancel the common factor.

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.5.4.2

Rewrite the expression.

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{1}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.5.5

Evaluate the exponent.

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.1.6

Multiply $4$ by $2$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (4 - 4 \sqrt{2} - 4 \sqrt{2} + 8) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.2

Add $4$ and $8$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (12 - 4 \sqrt{2} - 4 \sqrt{2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.7.3

Subtract $4 \sqrt{2}$ from $- 4 \sqrt{2}$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 6 (12 - 8 \sqrt{2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.8

Apply the distributive property.

$\frac{2 (- 56 + 40 \sqrt{2} + 6 \cdot 12 + 6 (- 8 \sqrt{2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.9

Multiply $6$ by $12$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 72 + 6 (- 8 \sqrt{2}) - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.10

Multiply $- 8$ by $6$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 72 - 48 \sqrt{2} - 12 (- 2 + 2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.11

Apply the distributive property.

$\frac{2 (- 56 + 40 \sqrt{2} + 72 - 48 \sqrt{2} - 12 \cdot - 2 - 12 (2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.12

Multiply $- 12$ by $- 2$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 72 - 48 \sqrt{2} + 24 - 12 (2 \sqrt{2}) - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.13

Multiply $2$ by $- 12$ .

$\frac{2 (- 56 + 40 \sqrt{2} + 72 - 48 \sqrt{2} + 24 - 24 \sqrt{2} - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.14

Add $- 56$ and $72$ .

$\frac{2 (16 + 40 \sqrt{2} - 48 \sqrt{2} + 24 - 24 \sqrt{2} - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.15

Add $16$ and $24$ .

$\frac{2 (40 + 40 \sqrt{2} - 48 \sqrt{2} - 24 \sqrt{2} - 8)}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.16

Subtract $8$ from $40$ .

$\frac{2 (32 + 40 \sqrt{2} - 48 \sqrt{2} - 24 \sqrt{2})}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.17

Subtract $48 \sqrt{2}$ from $40 \sqrt{2}$ .

$\frac{2 (32 - 8 \sqrt{2} - 24 \sqrt{2})}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.1.18

Subtract $24 \sqrt{2}$ from $- 8 \sqrt{2}$ .

$\frac{2 (32 - 32 \sqrt{2})}{{({(- 2 + 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$\frac{2 (32 - 32 \sqrt{2})}{{((- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2}) + 4)}^{3}}$

Step 9.2.2

Expand $(- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2})$ using the FOIL Method.

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Step 9.2.2.1

Apply the distributive property.

$\frac{2 (32 - 32 \sqrt{2})}{{(- 2 (- 2 + 2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2}) + 4)}^{3}}$

Step 9.2.2.2

Apply the distributive property.

$\frac{2 (32 - 32 \sqrt{2})}{{(- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2}) + 4)}^{3}}$

Step 9.2.2.3

Apply the distributive property.

$\frac{2 (32 - 32 \sqrt{2})}{{(- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) + 4)}^{3}}$

Step 9.2.3

Simplify and combine like terms.

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Step 9.2.3.1

Simplify each term.

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Step 9.2.3.1.1

Multiply $- 2$ by $- 2$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) + 4)}^{3}}$

Step 9.2.3.1.2

Multiply $2$ by $- 2$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) + 4)}^{3}}$

Step 9.2.3.1.3

Multiply $- 2$ by $2$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 2 \sqrt{2} (2 \sqrt{2}) + 4)}^{3}}$

Step 9.2.3.1.4

Multiply $2 \sqrt{2} (2 \sqrt{2})$ .

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Step 9.2.3.1.4.1

Multiply $2$ by $2$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2} + 4)}^{3}}$

Step 9.2.3.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} \sqrt{2}) + 4)}^{3}}$

Step 9.2.3.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 4)}^{3}}$

Step 9.2.3.1.4.4

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

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1} + 4)}^{3}}$

Step 9.2.3.1.4.5

Add $1$ and $1$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {\sqrt{2}}^{2} + 4)}^{3}}$

Step 9.2.3.1.5

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

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Step 9.2.3.1.5.1

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

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2} + 4)}^{3}}$

Step 9.2.3.1.5.2

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

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2} + 4)}^{3}}$

Step 9.2.3.1.5.3

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

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4)}^{3}}$

Step 9.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 9.2.3.1.5.4.1

Cancel the common factor.

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4)}^{3}}$

Step 9.2.3.1.5.4.2

Rewrite the expression.

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{1} + 4)}^{3}}$

Step 9.2.3.1.5.5

Evaluate the exponent.

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2 + 4)}^{3}}$

Step 9.2.3.1.6

Multiply $4$ by $2$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(4 - 4 \sqrt{2} - 4 \sqrt{2} + 8 + 4)}^{3}}$

Step 9.2.3.2

Add $4$ and $8$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(12 - 4 \sqrt{2} - 4 \sqrt{2} + 4)}^{3}}$

Step 9.2.3.3

Subtract $4 \sqrt{2}$ from $- 4 \sqrt{2}$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(12 - 8 \sqrt{2} + 4)}^{3}}$

Step 9.2.4

Add $12$ and $4$ .

$\frac{2 (32 - 32 \sqrt{2})}{{(16 - 8 \sqrt{2})}^{3}}$

Step 9.3

Use the Binomial Theorem.

$\frac{2 (32 - 32 \sqrt{2})}{16^{3} + 3 \cdot 16^{2} (- 8 \sqrt{2}) + 3 \cdot 16 {(- 8 \sqrt{2})}^{2} + {(- 8 \sqrt{2})}^{3}}$

Step 9.4

Simplify terms.

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Step 9.4.1

Simplify each term.

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Step 9.4.1.1

Raise $16$ to the power of $3$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 + 3 \cdot 16^{2} (- 8 \sqrt{2}) + 3 \cdot 16 {(- 8 \sqrt{2})}^{2} + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.2

Raise $16$ to the power of $2$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 + 3 \cdot 256 (- 8 \sqrt{2}) + 3 \cdot 16 {(- 8 \sqrt{2})}^{2} + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.3

Multiply $3$ by $256$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 + 768 (- 8 \sqrt{2}) + 3 \cdot 16 {(- 8 \sqrt{2})}^{2} + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.4

Multiply $- 8$ by $768$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 3 \cdot 16 {(- 8 \sqrt{2})}^{2} + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.5

Multiply $3$ by $16$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 {(- 8 \sqrt{2})}^{2} + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.6

Apply the product rule to $- 8 \sqrt{2}$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 ({(- 8)}^{2} {\sqrt{2}}^{2}) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.7

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 (64 {\sqrt{2}}^{2}) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.8

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

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Step 9.4.1.8.1

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 (64 {(2^{\frac{1}{2}})}^{2}) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.8.2

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 (64 \cdot 2^{\frac{1}{2} \cdot 2}) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.8.3

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 (64 \cdot 2^{\frac{2}{2}}) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.8.4

Cancel the common factor of $2$ .

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Step 9.4.1.8.4.1

Cancel the common factor.

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 (64 \cdot 2^{\frac{2}{2}}) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.8.4.2

Rewrite the expression.

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 (64 \cdot 2^{1}) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.8.5

Evaluate the exponent.

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 (64 \cdot 2) + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.9

Multiply $48 (64 \cdot 2)$ .

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Step 9.4.1.9.1

Multiply $64$ by $2$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 48 \cdot 128 + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.9.2

Multiply $48$ by $128$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 + {(- 8 \sqrt{2})}^{3}}$

Step 9.4.1.10

Apply the product rule to $- 8 \sqrt{2}$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 + {(- 8)}^{3} {\sqrt{2}}^{3}}$

Step 9.4.1.11

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 - 512 {\sqrt{2}}^{3}}$

Step 9.4.1.12

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 - 512 \sqrt{2^{3}}}$

Step 9.4.1.13

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 - 512 \sqrt{8}}$

Step 9.4.1.14

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 9.4.1.14.1

Factor $4$ out of $8$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 - 512 \sqrt{4 (2)}}$

Step 9.4.1.14.2

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

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 - 512 \sqrt{2^{2} \cdot 2}}$

Step 9.4.1.15

Pull terms out from under the radical.

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 - 512 (2 \sqrt{2})}$

Step 9.4.1.16

Multiply $2$ by $- 512$ .

$\frac{2 (32 - 32 \sqrt{2})}{4096 - 6144 \sqrt{2} + 6144 - 1024 \sqrt{2}}$

Step 9.4.2

Simplify by adding terms.

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Step 9.4.2.1

Add $4096$ and $6144$ .

$\frac{2 (32 - 32 \sqrt{2})}{10240 - 6144 \sqrt{2} - 1024 \sqrt{2}}$

Step 9.4.2.2

Subtract $1024 \sqrt{2}$ from $- 6144 \sqrt{2}$ .

$\frac{2 (32 - 32 \sqrt{2})}{10240 - 7168 \sqrt{2}}$

Step 9.5

Cancel the common factors.

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Step 9.5.1

Factor $2$ out of $10240$ .

$\frac{2 (32 - 32 \sqrt{2})}{2 \cdot 5120 - 7168 \sqrt{2}}$

Step 9.5.2

Factor $2$ out of $- 7168 \sqrt{2}$ .

$\frac{2 (32 - 32 \sqrt{2})}{2 \cdot 5120 + 2 (- 3584 \sqrt{2})}$

Step 9.5.3

Factor $2$ out of $2 (5120) + 2 (- 3584 \sqrt{2})$ .

$\frac{2 (32 - 32 \sqrt{2})}{2 (5120 - 3584 \sqrt{2})}$

Step 9.5.4

Cancel the common factor.

$\frac{2 (32 - 32 \sqrt{2})}{2 (5120 - 3584 \sqrt{2})}$

Step 9.5.5

Rewrite the expression.

$\frac{32 - 32 \sqrt{2}}{5120 - 3584 \sqrt{2}}$

Step 9.6

Cancel the common factor of $32 - 32 \sqrt{2}$ and $5120 - 3584 \sqrt{2}$ .

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Step 9.6.1

Factor $32$ out of $32$ .

$\frac{32 (1) - 32 \sqrt{2}}{5120 - 3584 \sqrt{2}}$

Step 9.6.2

Factor $32$ out of $- 32 \sqrt{2}$ .

$\frac{32 (1) + 32 (- \sqrt{2})}{5120 - 3584 \sqrt{2}}$

Step 9.6.3

Factor $32$ out of $32 (1) + 32 (- \sqrt{2})$ .

$\frac{32 (1 - \sqrt{2})}{5120 - 3584 \sqrt{2}}$

Step 9.6.4

Cancel the common factors.

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Step 9.6.4.1

Factor $32$ out of $5120$ .

$\frac{32 (1 - \sqrt{2})}{32 \cdot 160 - 3584 \sqrt{2}}$

Step 9.6.4.2

Factor $32$ out of $- 3584 \sqrt{2}$ .

$\frac{32 (1 - \sqrt{2})}{32 \cdot 160 + 32 (- 112 \sqrt{2})}$

Step 9.6.4.3

Factor $32$ out of $32 (160) + 32 (- 112 \sqrt{2})$ .

$\frac{32 (1 - \sqrt{2})}{32 (160 - 112 \sqrt{2})}$

Step 9.6.4.4

Cancel the common factor.

$\frac{32 (1 - \sqrt{2})}{32 (160 - 112 \sqrt{2})}$

Step 9.6.4.5

Rewrite the expression.

$\frac{1 - \sqrt{2}}{160 - 112 \sqrt{2}}$

Step 9.7

Multiply $\frac{1 - \sqrt{2}}{160 - 112 \sqrt{2}}$ by $\frac{160 + 112 \sqrt{2}}{160 + 112 \sqrt{2}}$ .

$\frac{1 - \sqrt{2}}{160 - 112 \sqrt{2}} \cdot \frac{160 + 112 \sqrt{2}}{160 + 112 \sqrt{2}}$

Step 9.8

Simplify terms.

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Step 9.8.1

Multiply $\frac{1 - \sqrt{2}}{160 - 112 \sqrt{2}}$ by $\frac{160 + 112 \sqrt{2}}{160 + 112 \sqrt{2}}$ .

$\frac{(1 - \sqrt{2}) (160 + 112 \sqrt{2})}{(160 - 112 \sqrt{2}) (160 + 112 \sqrt{2})}$

Step 9.8.2

Expand the denominator using the FOIL method.

$\frac{(1 - \sqrt{2}) (160 + 112 \sqrt{2})}{25600 + 17920 \sqrt{2} - 17920 \sqrt{2} - 12544 {\sqrt{2}}^{2}}$

Step 9.8.3

Simplify.

$\frac{(1 - \sqrt{2}) (160 + 112 \sqrt{2})}{512}$

Step 9.8.4

Cancel the common factor of $160 + 112 \sqrt{2}$ and $512$ .

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Step 9.8.4.1

Factor $16$ out of $(1 - \sqrt{2}) (160 + 112 \sqrt{2})$ .

$\frac{16 ((1 - \sqrt{2}) (10 + 7 \sqrt{2}))}{512}$

Step 9.8.4.2

Cancel the common factors.

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Step 9.8.4.2.1

Factor $16$ out of $512$ .

$\frac{16 ((1 - \sqrt{2}) (10 + 7 \sqrt{2}))}{16 (32)}$

Step 9.8.4.2.2

Cancel the common factor.

$\frac{16 ((1 - \sqrt{2}) (10 + 7 \sqrt{2}))}{16 \cdot 32}$

Step 9.8.4.2.3

Rewrite the expression.

$\frac{(1 - \sqrt{2}) (10 + 7 \sqrt{2})}{32}$

Step 9.9

Expand $(1 - \sqrt{2}) (10 + 7 \sqrt{2})$ using the FOIL Method.

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Step 9.9.1

Apply the distributive property.

$\frac{1 (10 + 7 \sqrt{2}) - \sqrt{2} (10 + 7 \sqrt{2})}{32}$

Step 9.9.2

Apply the distributive property.

$\frac{1 \cdot 10 + 1 (7 \sqrt{2}) - \sqrt{2} (10 + 7 \sqrt{2})}{32}$

Step 9.9.3

Apply the distributive property.

$\frac{1 \cdot 10 + 1 (7 \sqrt{2}) - \sqrt{2} \cdot 10 - \sqrt{2} (7 \sqrt{2})}{32}$

Step 9.10

Simplify and combine like terms.

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Step 9.10.1

Simplify each term.

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Step 9.10.1.1

Multiply $10$ by $1$ .

$\frac{10 + 1 (7 \sqrt{2}) - \sqrt{2} \cdot 10 - \sqrt{2} (7 \sqrt{2})}{32}$

Step 9.10.1.2

Multiply $7 \sqrt{2}$ by $1$ .

$\frac{10 + 7 \sqrt{2} - \sqrt{2} \cdot 10 - \sqrt{2} (7 \sqrt{2})}{32}$

Step 9.10.1.3

Multiply $10$ by $- 1$ .

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - \sqrt{2} (7 \sqrt{2})}{32}$

Step 9.10.1.4

Multiply $- \sqrt{2} (7 \sqrt{2})$ .

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Step 9.10.1.4.1

Multiply $7$ by $- 1$ .

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 \sqrt{2} \sqrt{2}}{32}$

Step 9.10.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 ({\sqrt{2}}^{1} \sqrt{2})}{32}$

Step 9.10.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{32}$

Step 9.10.1.4.4

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

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 {\sqrt{2}}^{1 + 1}}{32}$

Step 9.10.1.4.5

Add $1$ and $1$ .

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 {\sqrt{2}}^{2}}{32}$

Step 9.10.1.5

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

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Step 9.10.1.5.1

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

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 {(2^{\frac{1}{2}})}^{2}}{32}$

Step 9.10.1.5.2

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

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 \cdot 2^{\frac{1}{2} \cdot 2}}{32}$

Step 9.10.1.5.3

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

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 \cdot 2^{\frac{2}{2}}}{32}$

Step 9.10.1.5.4

Cancel the common factor of $2$ .

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Step 9.10.1.5.4.1

Cancel the common factor.

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 \cdot 2^{\frac{2}{2}}}{32}$

Step 9.10.1.5.4.2

Rewrite the expression.

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 \cdot 2^{1}}{32}$

Step 9.10.1.5.5

Evaluate the exponent.

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 7 \cdot 2}{32}$

Step 9.10.1.6

Multiply $- 7$ by $2$ .

$\frac{10 + 7 \sqrt{2} - 10 \sqrt{2} - 14}{32}$

Step 9.10.2

Subtract $14$ from $10$ .

$\frac{- 4 + 7 \sqrt{2} - 10 \sqrt{2}}{32}$

Step 9.10.3

Subtract $10 \sqrt{2}$ from $7 \sqrt{2}$ .

$\frac{- 4 - 3 \sqrt{2}}{32}$

Step 9.11

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) - 3 \sqrt{2}}{32}$

Step 9.12

Factor $- 1$ out of $- 3 \sqrt{2}$ .

$\frac{- 1 (4) - (3 \sqrt{2})}{32}$

Step 9.13

Factor $- 1$ out of $- 1 (4) - (3 \sqrt{2})$ .

$\frac{- 1 (4 + 3 \sqrt{2})}{32}$

Step 9.14

Move the negative in front of the fraction.

$- \frac{4 + 3 \sqrt{2}}{32}$

Step 10

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

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

Step 11

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

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Step 11.1

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

$f (- 2 + 2 \sqrt{2}) = \frac{(- 2 + 2 \sqrt{2}) + 2}{{(- 2 + 2 \sqrt{2})}^{2} + 4}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the numerator.

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Step 11.2.1.1

Add $- 2$ and $2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{0 + 2 \sqrt{2}}{{(- 2 + 2 \sqrt{2})}^{2} + 4}$

Step 11.2.1.2

Add $0$ and $2 \sqrt{2}$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{{(- 2 + 2 \sqrt{2})}^{2} + 4}$

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

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

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{(- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2}) + 4}$

Step 11.2.2.2

Expand $(- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2})$ using the FOIL Method.

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Step 11.2.2.2.1

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{- 2 (- 2 + 2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2}) + 4}$

Step 11.2.2.2.2

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2}) + 4}$

Step 11.2.2.2.3

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) + 4}$

Step 11.2.2.3

Simplify and combine like terms.

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Step 11.2.2.3.1

Simplify each term.

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Step 11.2.2.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) + 4}$

Step 11.2.2.3.1.2

Multiply $2$ by $- 2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) + 4}$

Step 11.2.2.3.1.3

Multiply $- 2$ by $2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 2 \sqrt{2} (2 \sqrt{2}) + 4}$

Step 11.2.2.3.1.4

Multiply $2 \sqrt{2} (2 \sqrt{2})$ .

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Step 11.2.2.3.1.4.1

Multiply $2$ by $2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2} + 4}$

Step 11.2.2.3.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) + 4}$

Step 11.2.2.3.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) + 4}$

Step 11.2.2.3.1.4.4

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

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

Step 11.2.2.3.1.4.5

Add $1$ and $1$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {\sqrt{2}}^{2} + 4}$

Step 11.2.2.3.1.5

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

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Step 11.2.2.3.1.5.1

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

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

Step 11.2.2.3.1.5.2

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

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

Step 11.2.2.3.1.5.3

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

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4}$

Step 11.2.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 11.2.2.3.1.5.4.1

Cancel the common factor.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4}$

Step 11.2.2.3.1.5.4.2

Rewrite the expression.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2 + 4}$

Step 11.2.2.3.1.5.5

Evaluate the exponent.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2 + 4}$

Step 11.2.2.3.1.6

Multiply $4$ by $2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{4 - 4 \sqrt{2} - 4 \sqrt{2} + 8 + 4}$

Step 11.2.2.3.2

Add $4$ and $8$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{12 - 4 \sqrt{2} - 4 \sqrt{2} + 4}$

Step 11.2.2.3.3

Subtract $4 \sqrt{2}$ from $- 4 \sqrt{2}$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{12 - 8 \sqrt{2} + 4}$

Step 11.2.2.4

Add $12$ and $4$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{16 - 8 \sqrt{2}}$

Step 11.2.3

Cancel the common factor of $2$ and $16 - 8 \sqrt{2}$ .

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Step 11.2.3.1

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

$f (- 2 + 2 \sqrt{2}) = \frac{2 (\sqrt{2})}{16 - 8 \sqrt{2}}$

Step 11.2.3.2

Cancel the common factors.

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Step 11.2.3.2.1

Factor $2$ out of $16$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{2 \cdot 8 - 8 \sqrt{2}}$

Step 11.2.3.2.2

Factor $2$ out of $- 8 \sqrt{2}$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{2 \cdot 8 + 2 (- 4 \sqrt{2})}$

Step 11.2.3.2.3

Factor $2$ out of $2 (8) + 2 (- 4 \sqrt{2})$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{2 (8 - 4 \sqrt{2})}$

Step 11.2.3.2.4

Cancel the common factor.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2}}{2 (8 - 4 \sqrt{2})}$

Step 11.2.3.2.5

Rewrite the expression.

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2}}{8 - 4 \sqrt{2}}$

Step 11.2.4

Multiply $\frac{\sqrt{2}}{8 - 4 \sqrt{2}}$ by $\frac{8 + 4 \sqrt{2}}{8 + 4 \sqrt{2}}$ .

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2}}{8 - 4 \sqrt{2}} \cdot \frac{8 + 4 \sqrt{2}}{8 + 4 \sqrt{2}}$

Step 11.2.5

Simplify terms.

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Step 11.2.5.1

Multiply $\frac{\sqrt{2}}{8 - 4 \sqrt{2}}$ by $\frac{8 + 4 \sqrt{2}}{8 + 4 \sqrt{2}}$ .

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2} (8 + 4 \sqrt{2})}{(8 - 4 \sqrt{2}) (8 + 4 \sqrt{2})}$

Step 11.2.5.2

Expand the denominator using the FOIL method.

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2} (8 + 4 \sqrt{2})}{64 + 32 \sqrt{2} - 32 \sqrt{2} - 16 {\sqrt{2}}^{2}}$

Step 11.2.5.3

Simplify.

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2} (8 + 4 \sqrt{2})}{32}$

Step 11.2.5.4

Cancel the common factor of $8 + 4 \sqrt{2}$ and $32$ .

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Step 11.2.5.4.1

Factor $4$ out of $\sqrt{2} (8 + 4 \sqrt{2})$ .

$f (- 2 + 2 \sqrt{2}) = \frac{4 (\sqrt{2} (2 + \sqrt{2}))}{32}$

Step 11.2.5.4.2

Cancel the common factors.

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Step 11.2.5.4.2.1

Factor $4$ out of $32$ .

$f (- 2 + 2 \sqrt{2}) = \frac{4 (\sqrt{2} (2 + \sqrt{2}))}{4 (8)}$

Step 11.2.5.4.2.2

Cancel the common factor.

$f (- 2 + 2 \sqrt{2}) = \frac{4 (\sqrt{2} (2 + \sqrt{2}))}{4 \cdot 8}$

Step 11.2.5.4.2.3

Rewrite the expression.

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2} (2 + \sqrt{2})}{8}$

Step 11.2.5.5

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2} \cdot 2 + \sqrt{2} \sqrt{2}}{8}$

Step 11.2.5.6

Move $2$ to the left of $\sqrt{2}$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}}{8}$

Step 11.2.5.7

Combine using the product rule for radicals.

$f (- 2 + 2 \sqrt{2}) = \frac{2 \cdot \sqrt{2} + \sqrt{2 \cdot 2}}{8}$

Step 11.2.6

Simplify each term.

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Step 11.2.6.1

Multiply $2$ by $2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2} + \sqrt{4}}{8}$

Step 11.2.6.2

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

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2} + \sqrt{2^{2}}}{8}$

Step 11.2.6.3

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

$f (- 2 + 2 \sqrt{2}) = \frac{2 \sqrt{2} + 2}{8}$

Step 11.2.7

Cancel the common factor of $2 \sqrt{2} + 2$ and $8$ .

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Step 11.2.7.1

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

$f (- 2 + 2 \sqrt{2}) = \frac{2 (\sqrt{2}) + 2}{8}$

Step 11.2.7.2

Factor $2$ out of $2$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 (\sqrt{2}) + 2 (1)}{8}$

Step 11.2.7.3

Factor $2$ out of $2 (\sqrt{2}) + 2 (1)$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 (\sqrt{2} + 1)}{8}$

Step 11.2.7.4

Cancel the common factors.

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Step 11.2.7.4.1

Factor $2$ out of $8$ .

$f (- 2 + 2 \sqrt{2}) = \frac{2 (\sqrt{2} + 1)}{2 (4)}$

Step 11.2.7.4.2

Cancel the common factor.

$f (- 2 + 2 \sqrt{2}) = \frac{2 (\sqrt{2} + 1)}{2 \cdot 4}$

Step 11.2.7.4.3

Rewrite the expression.

$f (- 2 + 2 \sqrt{2}) = \frac{\sqrt{2} + 1}{4}$

Step 11.2.8

The final answer is $\frac{\sqrt{2} + 1}{4}$ .

$y = \frac{\sqrt{2} + 1}{4}$

Step 12

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

$\frac{2 ({(- 2 - 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Use the Binomial Theorem.

$\frac{2 ({(- 2)}^{3} + 3 {(- 2)}^{2} (- 2 \sqrt{2}) + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2

Simplify each term.

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Step 13.1.2.1

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

$\frac{2 (- 8 + 3 {(- 2)}^{2} (- 2 \sqrt{2}) + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.2

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

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Step 13.1.2.2.1

Move $- 2$ .

$\frac{2 (- 8 + 3 (- 2 {(- 2)}^{2}) \sqrt{2} + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.2.2

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

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Step 13.1.2.2.2.1

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

$\frac{2 (- 8 + 3 ({(- 2)}^{1} {(- 2)}^{2}) \sqrt{2} + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.2.2.2

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

$\frac{2 (- 8 + 3 {(- 2)}^{1 + 2} \sqrt{2} + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.2.3

Add $1$ and $2$ .

$\frac{2 (- 8 + 3 {(- 2)}^{3} \sqrt{2} + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.3

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

$\frac{2 (- 8 + 3 \cdot - 8 \sqrt{2} + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.4

Multiply $3$ by $- 8$ .

$\frac{2 (- 8 - 24 \sqrt{2} + 3 \cdot - 2 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.5

Multiply $3$ by $- 2$ .

$\frac{2 (- 8 - 24 \sqrt{2} - 6 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.6

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

$\frac{2 (- 8 - 24 \sqrt{2} - 6 ({(- 2)}^{2} {\sqrt{2}}^{2}) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.7

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

$\frac{2 (- 8 - 24 \sqrt{2} - 6 (4 {\sqrt{2}}^{2}) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.8

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

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Step 13.1.2.8.1

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

$\frac{2 (- 8 - 24 \sqrt{2} - 6 (4 {(2^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.8.2

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

$\frac{2 (- 8 - 24 \sqrt{2} - 6 (4 \cdot 2^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.8.3

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

$\frac{2 (- 8 - 24 \sqrt{2} - 6 (4 \cdot 2^{\frac{2}{2}}) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.8.4

Cancel the common factor of $2$ .

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Step 13.1.2.8.4.1

Cancel the common factor.

$\frac{2 (- 8 - 24 \sqrt{2} - 6 (4 \cdot 2^{\frac{2}{2}}) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.8.4.2

Rewrite the expression.

$\frac{2 (- 8 - 24 \sqrt{2} - 6 (4 \cdot 2^{1}) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.8.5

Evaluate the exponent.

$\frac{2 (- 8 - 24 \sqrt{2} - 6 (4 \cdot 2) + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.9

Multiply $- 6 (4 \cdot 2)$ .

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Step 13.1.2.9.1

Multiply $4$ by $2$ .

$\frac{2 (- 8 - 24 \sqrt{2} - 6 \cdot 8 + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.9.2

Multiply $- 6$ by $8$ .

$\frac{2 (- 8 - 24 \sqrt{2} - 48 + {(- 2 \sqrt{2})}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.10

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

$\frac{2 (- 8 - 24 \sqrt{2} - 48 + {(- 2)}^{3} {\sqrt{2}}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.11

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

$\frac{2 (- 8 - 24 \sqrt{2} - 48 - 8 {\sqrt{2}}^{3} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.12

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

$\frac{2 (- 8 - 24 \sqrt{2} - 48 - 8 \sqrt{2^{3}} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.13

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

$\frac{2 (- 8 - 24 \sqrt{2} - 48 - 8 \sqrt{8} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.14

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 13.1.2.14.1

Factor $4$ out of $8$ .

$\frac{2 (- 8 - 24 \sqrt{2} - 48 - 8 \sqrt{4 (2)} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.14.2

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

$\frac{2 (- 8 - 24 \sqrt{2} - 48 - 8 \sqrt{2^{2} \cdot 2} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.15

Pull terms out from under the radical.

$\frac{2 (- 8 - 24 \sqrt{2} - 48 - 8 (2 \sqrt{2}) + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.2.16

Multiply $2$ by $- 8$ .

$\frac{2 (- 8 - 24 \sqrt{2} - 48 - 16 \sqrt{2} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.3

Subtract $48$ from $- 8$ .

$\frac{2 (- 56 - 24 \sqrt{2} - 16 \sqrt{2} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.4

Subtract $16 \sqrt{2}$ from $- 24 \sqrt{2}$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 {(- 2 - 2 \sqrt{2})}^{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.5

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

$\frac{2 (- 56 - 40 \sqrt{2} + 6 ((- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.6

Expand $(- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2})$ using the FOIL Method.

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Step 13.1.6.1

Apply the distributive property.

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (- 2 (- 2 - 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.6.2

Apply the distributive property.

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.6.3

Apply the distributive property.

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7

Simplify and combine like terms.

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Step 13.1.7.1

Simplify each term.

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Step 13.1.7.1.1

Multiply $- 2$ by $- 2$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.2

Multiply $- 2$ by $- 2$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.3

Multiply $- 2$ by $- 2$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} - 2 \sqrt{2} (- 2 \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.4

Multiply $- 2 \sqrt{2} (- 2 \sqrt{2})$ .

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Step 13.1.7.1.4.1

Multiply $- 2$ by $- 2$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} \sqrt{2})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.4.4

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

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.4.5

Add $1$ and $1$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {\sqrt{2}}^{2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.5

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

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Step 13.1.7.1.5.1

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

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.5.2

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

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.5.3

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

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 13.1.7.1.5.4.1

Cancel the common factor.

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.5.4.2

Rewrite the expression.

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{1}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.5.5

Evaluate the exponent.

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.1.6

Multiply $4$ by $2$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (4 + 4 \sqrt{2} + 4 \sqrt{2} + 8) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.2

Add $4$ and $8$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (12 + 4 \sqrt{2} + 4 \sqrt{2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.7.3

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 6 (12 + 8 \sqrt{2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.8

Apply the distributive property.

$\frac{2 (- 56 - 40 \sqrt{2} + 6 \cdot 12 + 6 (8 \sqrt{2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.9

Multiply $6$ by $12$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 72 + 6 (8 \sqrt{2}) - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.10

Multiply $8$ by $6$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 72 + 48 \sqrt{2} - 12 (- 2 - 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.11

Apply the distributive property.

$\frac{2 (- 56 - 40 \sqrt{2} + 72 + 48 \sqrt{2} - 12 \cdot - 2 - 12 (- 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.12

Multiply $- 12$ by $- 2$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 72 + 48 \sqrt{2} + 24 - 12 (- 2 \sqrt{2}) - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.13

Multiply $- 2$ by $- 12$ .

$\frac{2 (- 56 - 40 \sqrt{2} + 72 + 48 \sqrt{2} + 24 + 24 \sqrt{2} - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.14

Add $- 56$ and $72$ .

$\frac{2 (16 - 40 \sqrt{2} + 48 \sqrt{2} + 24 + 24 \sqrt{2} - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.15

Add $16$ and $24$ .

$\frac{2 (40 - 40 \sqrt{2} + 48 \sqrt{2} + 24 \sqrt{2} - 8)}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.16

Subtract $8$ from $40$ .

$\frac{2 (32 - 40 \sqrt{2} + 48 \sqrt{2} + 24 \sqrt{2})}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.17

Add $- 40 \sqrt{2}$ and $48 \sqrt{2}$ .

$\frac{2 (32 + 8 \sqrt{2} + 24 \sqrt{2})}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.1.18

Add $8 \sqrt{2}$ and $24 \sqrt{2}$ .

$\frac{2 (32 + 32 \sqrt{2})}{{({(- 2 - 2 \sqrt{2})}^{2} + 4)}^{3}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{2 (32 + 32 \sqrt{2})}{{((- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2}) + 4)}^{3}}$

Step 13.2.2

Expand $(- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2})$ using the FOIL Method.

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Step 13.2.2.1

Apply the distributive property.

$\frac{2 (32 + 32 \sqrt{2})}{{(- 2 (- 2 - 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2}) + 4)}^{3}}$

Step 13.2.2.2

Apply the distributive property.

$\frac{2 (32 + 32 \sqrt{2})}{{(- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2}) + 4)}^{3}}$

Step 13.2.2.3

Apply the distributive property.

$\frac{2 (32 + 32 \sqrt{2})}{{(- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) + 4)}^{3}}$

Step 13.2.3

Simplify and combine like terms.

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Step 13.2.3.1

Simplify each term.

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Step 13.2.3.1.1

Multiply $- 2$ by $- 2$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) + 4)}^{3}}$

Step 13.2.3.1.2

Multiply $- 2$ by $- 2$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) + 4)}^{3}}$

Step 13.2.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} - 2 \sqrt{2} (- 2 \sqrt{2}) + 4)}^{3}}$

Step 13.2.3.1.4

Multiply $- 2 \sqrt{2} (- 2 \sqrt{2})$ .

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Step 13.2.3.1.4.1

Multiply $- 2$ by $- 2$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2} + 4)}^{3}}$

Step 13.2.3.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} \sqrt{2}) + 4)}^{3}}$

Step 13.2.3.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 4)}^{3}}$

Step 13.2.3.1.4.4

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

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1} + 4)}^{3}}$

Step 13.2.3.1.4.5

Add $1$ and $1$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {\sqrt{2}}^{2} + 4)}^{3}}$

Step 13.2.3.1.5

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

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Step 13.2.3.1.5.1

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

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2} + 4)}^{3}}$

Step 13.2.3.1.5.2

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

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2} + 4)}^{3}}$

Step 13.2.3.1.5.3

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

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4)}^{3}}$

Step 13.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 13.2.3.1.5.4.1

Cancel the common factor.

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4)}^{3}}$

Step 13.2.3.1.5.4.2

Rewrite the expression.

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{1} + 4)}^{3}}$

Step 13.2.3.1.5.5

Evaluate the exponent.

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2 + 4)}^{3}}$

Step 13.2.3.1.6

Multiply $4$ by $2$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(4 + 4 \sqrt{2} + 4 \sqrt{2} + 8 + 4)}^{3}}$

Step 13.2.3.2

Add $4$ and $8$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(12 + 4 \sqrt{2} + 4 \sqrt{2} + 4)}^{3}}$

Step 13.2.3.3

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(12 + 8 \sqrt{2} + 4)}^{3}}$

Step 13.2.4

Add $12$ and $4$ .

$\frac{2 (32 + 32 \sqrt{2})}{{(16 + 8 \sqrt{2})}^{3}}$

Step 13.3

Use the Binomial Theorem.

$\frac{2 (32 + 32 \sqrt{2})}{16^{3} + 3 \cdot 16^{2} (8 \sqrt{2}) + 3 \cdot 16 {(8 \sqrt{2})}^{2} + {(8 \sqrt{2})}^{3}}$

Step 13.4

Simplify terms.

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Step 13.4.1

Simplify each term.

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Step 13.4.1.1

Raise $16$ to the power of $3$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 3 \cdot 16^{2} (8 \sqrt{2}) + 3 \cdot 16 {(8 \sqrt{2})}^{2} + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.2

Raise $16$ to the power of $2$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 3 \cdot 256 (8 \sqrt{2}) + 3 \cdot 16 {(8 \sqrt{2})}^{2} + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.3

Multiply $3$ by $256$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 768 (8 \sqrt{2}) + 3 \cdot 16 {(8 \sqrt{2})}^{2} + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.4

Multiply $8$ by $768$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 3 \cdot 16 {(8 \sqrt{2})}^{2} + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.5

Multiply $3$ by $16$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 {(8 \sqrt{2})}^{2} + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.6

Apply the product rule to $8 \sqrt{2}$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (8^{2} {\sqrt{2}}^{2}) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.7

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (64 {\sqrt{2}}^{2}) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.8

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

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Step 13.4.1.8.1

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (64 {(2^{\frac{1}{2}})}^{2}) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.8.2

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (64 \cdot 2^{\frac{1}{2} \cdot 2}) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.8.3

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (64 \cdot 2^{\frac{2}{2}}) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.8.4

Cancel the common factor of $2$ .

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Step 13.4.1.8.4.1

Cancel the common factor.

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (64 \cdot 2^{\frac{2}{2}}) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.8.4.2

Rewrite the expression.

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (64 \cdot 2^{1}) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.8.5

Evaluate the exponent.

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 (64 \cdot 2) + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.9

Multiply $48 (64 \cdot 2)$ .

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Step 13.4.1.9.1

Multiply $64$ by $2$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 48 \cdot 128 + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.9.2

Multiply $48$ by $128$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + {(8 \sqrt{2})}^{3}}$

Step 13.4.1.10

Apply the product rule to $8 \sqrt{2}$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 8^{3} {\sqrt{2}}^{3}}$

Step 13.4.1.11

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 512 {\sqrt{2}}^{3}}$

Step 13.4.1.12

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 512 \sqrt{2^{3}}}$

Step 13.4.1.13

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 512 \sqrt{8}}$

Step 13.4.1.14

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 13.4.1.14.1

Factor $4$ out of $8$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 512 \sqrt{4 (2)}}$

Step 13.4.1.14.2

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

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 512 \sqrt{2^{2} \cdot 2}}$

Step 13.4.1.15

Pull terms out from under the radical.

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 512 (2 \sqrt{2})}$

Step 13.4.1.16

Multiply $2$ by $512$ .

$\frac{2 (32 + 32 \sqrt{2})}{4096 + 6144 \sqrt{2} + 6144 + 1024 \sqrt{2}}$

Step 13.4.2

Simplify by adding terms.

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Step 13.4.2.1

Add $4096$ and $6144$ .

$\frac{2 (32 + 32 \sqrt{2})}{10240 + 6144 \sqrt{2} + 1024 \sqrt{2}}$

Step 13.4.2.2

Add $6144 \sqrt{2}$ and $1024 \sqrt{2}$ .

$\frac{2 (32 + 32 \sqrt{2})}{10240 + 7168 \sqrt{2}}$

Step 13.5

Cancel the common factors.

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Step 13.5.1

Factor $2$ out of $10240$ .

$\frac{2 (32 + 32 \sqrt{2})}{2 \cdot 5120 + 7168 \sqrt{2}}$

Step 13.5.2

Factor $2$ out of $7168 \sqrt{2}$ .

$\frac{2 (32 + 32 \sqrt{2})}{2 \cdot 5120 + 2 (3584 \sqrt{2})}$

Step 13.5.3

Factor $2$ out of $2 (5120) + 2 (3584 \sqrt{2})$ .

$\frac{2 (32 + 32 \sqrt{2})}{2 (5120 + 3584 \sqrt{2})}$

Step 13.5.4

Cancel the common factor.

$\frac{2 (32 + 32 \sqrt{2})}{2 (5120 + 3584 \sqrt{2})}$

Step 13.5.5

Rewrite the expression.

$\frac{32 + 32 \sqrt{2}}{5120 + 3584 \sqrt{2}}$

Step 13.6

Cancel the common factor of $32 + 32 \sqrt{2}$ and $5120 + 3584 \sqrt{2}$ .

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Step 13.6.1

Factor $32$ out of $32$ .

$\frac{32 \cdot 1 + 32 \sqrt{2}}{5120 + 3584 \sqrt{2}}$

Step 13.6.2

Factor $32$ out of $32 \sqrt{2}$ .

$\frac{32 \cdot 1 + 32 (\sqrt{2})}{5120 + 3584 \sqrt{2}}$

Step 13.6.3

Factor $32$ out of $32 \cdot 1 + 32 (\sqrt{2})$ .

$\frac{32 \cdot (1 + \sqrt{2})}{5120 + 3584 \sqrt{2}}$

Step 13.6.4

Cancel the common factors.

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Step 13.6.4.1

Factor $32$ out of $5120$ .

$\frac{32 \cdot (1 + \sqrt{2})}{32 (160) + 3584 \sqrt{2}}$

Step 13.6.4.2

Factor $32$ out of $3584 \sqrt{2}$ .

$\frac{32 \cdot (1 + \sqrt{2})}{32 (160) + 32 (112 \sqrt{2})}$

Step 13.6.4.3

Factor $32$ out of $32 (160) + 32 (112 \sqrt{2})$ .

$\frac{32 \cdot (1 + \sqrt{2})}{32 (160 + 112 \sqrt{2})}$

Step 13.6.4.4

Cancel the common factor.

$\frac{32 \cdot (1 + \sqrt{2})}{32 (160 + 112 \sqrt{2})}$

Step 13.6.4.5

Rewrite the expression.

$\frac{1 + \sqrt{2}}{160 + 112 \sqrt{2}}$

Step 13.7

Multiply $\frac{1 + \sqrt{2}}{160 + 112 \sqrt{2}}$ by $\frac{160 - 112 \sqrt{2}}{160 - 112 \sqrt{2}}$ .

$\frac{1 + \sqrt{2}}{160 + 112 \sqrt{2}} \cdot \frac{160 - 112 \sqrt{2}}{160 - 112 \sqrt{2}}$

Step 13.8

Simplify terms.

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Step 13.8.1

Multiply $\frac{1 + \sqrt{2}}{160 + 112 \sqrt{2}}$ by $\frac{160 - 112 \sqrt{2}}{160 - 112 \sqrt{2}}$ .

$\frac{(1 + \sqrt{2}) (160 - 112 \sqrt{2})}{(160 + 112 \sqrt{2}) (160 - 112 \sqrt{2})}$

Step 13.8.2

Expand the denominator using the FOIL method.

$\frac{(1 + \sqrt{2}) (160 - 112 \sqrt{2})}{25600 - 17920 \sqrt{2} + 17920 \sqrt{2} - 12544 {\sqrt{2}}^{2}}$

Step 13.8.3

Simplify.

$\frac{(1 + \sqrt{2}) (160 - 112 \sqrt{2})}{512}$

Step 13.8.4

Cancel the common factor of $160 - 112 \sqrt{2}$ and $512$ .

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Step 13.8.4.1

Factor $16$ out of $(1 + \sqrt{2}) (160 - 112 \sqrt{2})$ .

$\frac{16 ((1 + \sqrt{2}) (10 - 7 \sqrt{2}))}{512}$

Step 13.8.4.2

Cancel the common factors.

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Step 13.8.4.2.1

Factor $16$ out of $512$ .

$\frac{16 ((1 + \sqrt{2}) (10 - 7 \sqrt{2}))}{16 (32)}$

Step 13.8.4.2.2

Cancel the common factor.

$\frac{16 ((1 + \sqrt{2}) (10 - 7 \sqrt{2}))}{16 \cdot 32}$

Step 13.8.4.2.3

Rewrite the expression.

$\frac{(1 + \sqrt{2}) (10 - 7 \sqrt{2})}{32}$

Step 13.9

Expand $(1 + \sqrt{2}) (10 - 7 \sqrt{2})$ using the FOIL Method.

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Step 13.9.1

Apply the distributive property.

$\frac{1 (10 - 7 \sqrt{2}) + \sqrt{2} (10 - 7 \sqrt{2})}{32}$

Step 13.9.2

Apply the distributive property.

$\frac{1 \cdot 10 + 1 (- 7 \sqrt{2}) + \sqrt{2} (10 - 7 \sqrt{2})}{32}$

Step 13.9.3

Apply the distributive property.

$\frac{1 \cdot 10 + 1 (- 7 \sqrt{2}) + \sqrt{2} \cdot 10 + \sqrt{2} (- 7 \sqrt{2})}{32}$

Step 13.10

Simplify and combine like terms.

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Step 13.10.1

Simplify each term.

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Step 13.10.1.1

Multiply $10$ by $1$ .

$\frac{10 + 1 (- 7 \sqrt{2}) + \sqrt{2} \cdot 10 + \sqrt{2} (- 7 \sqrt{2})}{32}$

Step 13.10.1.2

Multiply $- 7 \sqrt{2}$ by $1$ .

$\frac{10 - 7 \sqrt{2} + \sqrt{2} \cdot 10 + \sqrt{2} (- 7 \sqrt{2})}{32}$

Step 13.10.1.3

Move $10$ to the left of $\sqrt{2}$ .

$\frac{10 - 7 \sqrt{2} + 10 \cdot \sqrt{2} + \sqrt{2} (- 7 \sqrt{2})}{32}$

Step 13.10.1.4

Multiply $\sqrt{2} (- 7 \sqrt{2})$ .

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Step 13.10.1.4.1

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 ({\sqrt{2}}^{1} \sqrt{2})}{32}$

Step 13.10.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{32}$

Step 13.10.1.4.3

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

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 {\sqrt{2}}^{1 + 1}}{32}$

Step 13.10.1.4.4

Add $1$ and $1$ .

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 {\sqrt{2}}^{2}}{32}$

Step 13.10.1.5

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

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Step 13.10.1.5.1

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

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 {(2^{\frac{1}{2}})}^{2}}{32}$

Step 13.10.1.5.2

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

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 \cdot 2^{\frac{1}{2} \cdot 2}}{32}$

Step 13.10.1.5.3

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

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 \cdot 2^{\frac{2}{2}}}{32}$

Step 13.10.1.5.4

Cancel the common factor of $2$ .

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Step 13.10.1.5.4.1

Cancel the common factor.

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 \cdot 2^{\frac{2}{2}}}{32}$

Step 13.10.1.5.4.2

Rewrite the expression.

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 \cdot 2^{1}}{32}$

Step 13.10.1.5.5

Evaluate the exponent.

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 7 \cdot 2}{32}$

Step 13.10.1.6

Multiply $- 7$ by $2$ .

$\frac{10 - 7 \sqrt{2} + 10 \sqrt{2} - 14}{32}$

Step 13.10.2

Subtract $14$ from $10$ .

$\frac{- 4 - 7 \sqrt{2} + 10 \sqrt{2}}{32}$

Step 13.10.3

Add $- 7 \sqrt{2}$ and $10 \sqrt{2}$ .

$\frac{- 4 + 3 \sqrt{2}}{32}$

Step 13.11

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) + 3 \sqrt{2}}{32}$

Step 13.12

Factor $- 1$ out of $3 \sqrt{2}$ .

$\frac{- 1 (4) - (- 3 \sqrt{2})}{32}$

Step 13.13

Factor $- 1$ out of $- 1 (4) - (- 3 \sqrt{2})$ .

$\frac{- 1 (4 - 3 \sqrt{2})}{32}$

Step 13.14

Move the negative in front of the fraction.

$- \frac{4 - 3 \sqrt{2}}{32}$

Step 14

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

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

Step 15

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

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Step 15.1

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

$f (- 2 - 2 \sqrt{2}) = \frac{(- 2 - 2 \sqrt{2}) + 2}{{(- 2 - 2 \sqrt{2})}^{2} + 4}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify the numerator.

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Step 15.2.1.1

Add $- 2$ and $2$ .

$f (- 2 - 2 \sqrt{2}) = \frac{0 - 2 \sqrt{2}}{{(- 2 - 2 \sqrt{2})}^{2} + 4}$

Step 15.2.1.2

Subtract $2 \sqrt{2}$ from $0$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{{(- 2 - 2 \sqrt{2})}^{2} + 4}$

Step 15.2.2

Simplify the denominator.

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Step 15.2.2.1

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

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{(- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2}) + 4}$

Step 15.2.2.2

Expand $(- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2})$ using the FOIL Method.

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Step 15.2.2.2.1

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{- 2 (- 2 - 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2}) + 4}$

Step 15.2.2.2.2

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2}) + 4}$

Step 15.2.2.2.3

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) + 4}$

Step 15.2.2.3

Simplify and combine like terms.

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Step 15.2.2.3.1

Simplify each term.

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Step 15.2.2.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) + 4}$

Step 15.2.2.3.1.2

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) + 4}$

Step 15.2.2.3.1.3

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} - 2 \sqrt{2} (- 2 \sqrt{2}) + 4}$

Step 15.2.2.3.1.4

Multiply $- 2 \sqrt{2} (- 2 \sqrt{2})$ .

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Step 15.2.2.3.1.4.1

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2} + 4}$

Step 15.2.2.3.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) + 4}$

Step 15.2.2.3.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) + 4}$

Step 15.2.2.3.1.4.4

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

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

Step 15.2.2.3.1.4.5

Add $1$ and $1$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {\sqrt{2}}^{2} + 4}$

Step 15.2.2.3.1.5

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

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Step 15.2.2.3.1.5.1

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

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

Step 15.2.2.3.1.5.2

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

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

Step 15.2.2.3.1.5.3

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

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4}$

Step 15.2.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 15.2.2.3.1.5.4.1

Cancel the common factor.

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} + 4}$

Step 15.2.2.3.1.5.4.2

Rewrite the expression.

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2 + 4}$

Step 15.2.2.3.1.5.5

Evaluate the exponent.

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2 + 4}$

Step 15.2.2.3.1.6

Multiply $4$ by $2$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{4 + 4 \sqrt{2} + 4 \sqrt{2} + 8 + 4}$

Step 15.2.2.3.2

Add $4$ and $8$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{12 + 4 \sqrt{2} + 4 \sqrt{2} + 4}$

Step 15.2.2.3.3

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{12 + 8 \sqrt{2} + 4}$

Step 15.2.2.4

Add $12$ and $4$ .

$f (- 2 - 2 \sqrt{2}) = \frac{- 2 \sqrt{2}}{16 + 8 \sqrt{2}}$

Step 15.2.3

Reduce the expression by cancelling the common factors.

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Step 15.2.3.1

Cancel the common factor of $- 2$ and $16 + 8 \sqrt{2}$ .

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Step 15.2.3.1.1

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

$f (- 2 - 2 \sqrt{2}) = \frac{2 (- \sqrt{2})}{16 + 8 \sqrt{2}}$

Step 15.2.3.1.2

Cancel the common factors.

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Step 15.2.3.1.2.1

Factor $2$ out of $16$ .

$f (- 2 - 2 \sqrt{2}) = \frac{2 (- \sqrt{2})}{2 (8) + 8 \sqrt{2}}$

Step 15.2.3.1.2.2

Factor $2$ out of $8 \sqrt{2}$ .

$f (- 2 - 2 \sqrt{2}) = \frac{2 (- \sqrt{2})}{2 (8) + 2 (4 \sqrt{2})}$

Step 15.2.3.1.2.3

Factor $2$ out of $2 (8) + 2 (4 \sqrt{2})$ .

$f (- 2 - 2 \sqrt{2}) = \frac{2 (- \sqrt{2})}{2 (8 + 4 \sqrt{2})}$

Step 15.2.3.1.2.4

Cancel the common factor.

$f (- 2 - 2 \sqrt{2}) = \frac{2 (- \sqrt{2})}{2 (8 + 4 \sqrt{2})}$

Step 15.2.3.1.2.5

Rewrite the expression.

$f (- 2 - 2 \sqrt{2}) = \frac{- \sqrt{2}}{8 + 4 \sqrt{2}}$

Step 15.2.3.2

Move the negative in front of the fraction.

$f (- 2 - 2 \sqrt{2}) = - \frac{\sqrt{2}}{8 + 4 \sqrt{2}}$

Step 15.2.4

Multiply $\frac{\sqrt{2}}{8 + 4 \sqrt{2}}$ by $\frac{8 - 4 \sqrt{2}}{8 - 4 \sqrt{2}}$ .

$f (- 2 - 2 \sqrt{2}) = - (\frac{\sqrt{2}}{8 + 4 \sqrt{2}} \cdot \frac{8 - 4 \sqrt{2}}{8 - 4 \sqrt{2}})$

Step 15.2.5

Multiply $\frac{\sqrt{2}}{8 + 4 \sqrt{2}}$ by $\frac{8 - 4 \sqrt{2}}{8 - 4 \sqrt{2}}$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{\sqrt{2} (8 - 4 \sqrt{2})}{(8 + 4 \sqrt{2}) (8 - 4 \sqrt{2})}$

Step 15.2.6

Expand the denominator using the FOIL method.

$f (- 2 - 2 \sqrt{2}) = - \frac{\sqrt{2} (8 - 4 \sqrt{2})}{64 - 32 \sqrt{2} + 32 \sqrt{2} - 16 {\sqrt{2}}^{2}}$

Step 15.2.7

Simplify.

$f (- 2 - 2 \sqrt{2}) = - \frac{\sqrt{2} (8 - 4 \sqrt{2})}{32}$

Step 15.2.8

Cancel the common factor of $8 - 4 \sqrt{2}$ and $32$ .

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Step 15.2.8.1

Factor $4$ out of $\sqrt{2} (8 - 4 \sqrt{2})$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{4 (\sqrt{2} (2 - \sqrt{2}))}{32}$

Step 15.2.8.2

Cancel the common factors.

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Step 15.2.8.2.1

Factor $4$ out of $32$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{4 (\sqrt{2} (2 - \sqrt{2}))}{4 (8)}$

Step 15.2.8.2.2

Cancel the common factor.

$f (- 2 - 2 \sqrt{2}) = - \frac{4 (\sqrt{2} (2 - \sqrt{2}))}{4 \cdot 8}$

Step 15.2.8.2.3

Rewrite the expression.

$f (- 2 - 2 \sqrt{2}) = - \frac{\sqrt{2} (2 - \sqrt{2})}{8}$

Step 15.2.9

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = - \frac{\sqrt{2} \cdot 2 + \sqrt{2} (- \sqrt{2})}{8}$

Step 15.2.10

Move $2$ to the left of $\sqrt{2}$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \cdot \sqrt{2} + \sqrt{2} (- \sqrt{2})}{8}$

Step 15.2.11

Multiply $\sqrt{2} (- \sqrt{2})$ .

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Step 15.2.11.1

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \cdot \sqrt{2} - (\sqrt{2} \sqrt{2})}{8}$

Step 15.2.11.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \cdot \sqrt{2} - (\sqrt{2} \sqrt{2})}{8}$

Step 15.2.11.3

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

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \cdot \sqrt{2} - {\sqrt{2}}^{1 + 1}}{8}$

Step 15.2.11.4

Add $1$ and $1$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \cdot \sqrt{2} - {\sqrt{2}}^{2}}{8}$

Step 15.2.12

Simplify each term.

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Step 15.2.12.1

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

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Step 15.2.12.1.1

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

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}{8}$

Step 15.2.12.1.2

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

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}{8}$

Step 15.2.12.1.3

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

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \sqrt{2} - 2^{\frac{2}{2}}}{8}$

Step 15.2.12.1.4

Cancel the common factor of $2$ .

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Step 15.2.12.1.4.1

Cancel the common factor.

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \sqrt{2} - 2^{\frac{2}{2}}}{8}$

Step 15.2.12.1.4.2

Rewrite the expression.

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \sqrt{2} - 2}{8}$

Step 15.2.12.1.5

Evaluate the exponent.

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \sqrt{2} - 1 \cdot 2}{8}$

Step 15.2.12.2

Multiply $- 1$ by $2$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 \sqrt{2} - 2}{8}$

Step 15.2.13

Cancel the common factor of $2 \sqrt{2} - 2$ and $8$ .

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Step 15.2.13.1

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

$f (- 2 - 2 \sqrt{2}) = - \frac{2 (\sqrt{2}) - 2}{8}$

Step 15.2.13.2

Factor $2$ out of $- 2$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 (\sqrt{2}) + 2 (- 1)}{8}$

Step 15.2.13.3

Factor $2$ out of $2 (\sqrt{2}) + 2 (- 1)$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 (\sqrt{2} - 1)}{8}$

Step 15.2.13.4

Cancel the common factors.

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Step 15.2.13.4.1

Factor $2$ out of $8$ .

$f (- 2 - 2 \sqrt{2}) = - \frac{2 (\sqrt{2} - 1)}{2 (4)}$

Step 15.2.13.4.2

Cancel the common factor.

$f (- 2 - 2 \sqrt{2}) = - \frac{2 (\sqrt{2} - 1)}{2 \cdot 4}$

Step 15.2.13.4.3

Rewrite the expression.

$f (- 2 - 2 \sqrt{2}) = - \frac{\sqrt{2} - 1}{4}$

Step 15.2.14

The final answer is $- \frac{\sqrt{2} - 1}{4}$ .

$y = - \frac{\sqrt{2} - 1}{4}$

Step 16

These are the local extrema for $f (x) = \frac{x + 2}{x^{2} + 4}$ .

$(- 2 + 2 \sqrt{2}, \frac{\sqrt{2} + 1}{4})$ is a local maxima

$(- 2 - 2 \sqrt{2}, - \frac{\sqrt{2} - 1}{4})$ is a local minima

Step 17