Calculus Examples

$lim_{x \to \frac{π}{4}} \frac{4 - 4 \tan (x)}{\sin (x) - \cos (x)}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \frac{π}{4}} 4 - 4 \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{lim_{x \to \frac{π}{4}} 4 - lim_{x \to \frac{π}{4}} 4 \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2.1.2

Evaluate the limit of $4$ which is constant as $x$ approaches $\frac{π}{4}$ .

$\frac{4 - lim_{x \to \frac{π}{4}} 4 \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2.1.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 - 4 lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2.1.4

Move the limit inside the trig function because tangent is continuous.

$\frac{4 - 4 \tan (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{4 - 4 \tan (\frac{π}{4})}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{4 - 4 \cdot 1}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2.3.1.2

Multiply $- 4$ by $1$ .

$\frac{4 - 4}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.2.3.2

Subtract $4$ from $4$ .

$\frac{0}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{0}{lim_{x \to \frac{π}{4}} \sin (x) - lim_{x \to \frac{π}{4}} \cos (x)}$

Step 1.3.2

Move the limit inside the trig function because sine is continuous.

$\frac{0}{\sin (lim_{x \to \frac{π}{4}} x) - lim_{x \to \frac{π}{4}} \cos (x)}$

Step 1.3.3

Move the limit inside the trig function because cosine is continuous.

$\frac{0}{\sin (lim_{x \to \frac{π}{4}} x) - \cos (lim_{x \to \frac{π}{4}} x)}$

Step 1.3.4

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 1.3.4.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{0}{\sin (\frac{π}{4}) - \cos (lim_{x \to \frac{π}{4}} x)}$

Step 1.3.4.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{0}{\sin (\frac{π}{4}) - \cos (\frac{π}{4})}$

Step 1.3.5

Simplify the answer.

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Step 1.3.5.1

Simplify each term.

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Step 1.3.5.1.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{0}{\frac{\sqrt{2}}{2} - \cos (\frac{π}{4})}$

Step 1.3.5.1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{0}{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}}$

Step 1.3.5.2

Combine the numerators over the common denominator.

$\frac{0}{\frac{\sqrt{2} - \sqrt{2}}{2}}$

Step 1.3.5.3

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

$\frac{0}{\frac{0}{2}}$

Step 1.3.5.4

Divide $0$ by $2$ .

$\frac{0}{0}$

Step 1.3.5.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.3.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \frac{π}{4}} \frac{4 - 4 \tan (x)}{\sin (x) - \cos (x)} = lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [4 - 4 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [4 - 4 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.2

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

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [4] + \frac{d}{d x} [- 4 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.3

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

$lim_{x \to \frac{π}{4}} \frac{0 + \frac{d}{d x} [- 4 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.4

Evaluate $\frac{d}{d x} [- 4 \tan (x)]$ .

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Step 3.4.1

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

$lim_{x \to \frac{π}{4}} \frac{0 - 4 \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.4.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to \frac{π}{4}} \frac{0 - 4 \sec^{2} (x)}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.5

Simplify.

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Step 3.5.1

Subtract $4 \sec^{2} (x)$ from $0$ .

$lim_{x \to \frac{π}{4}} \frac{- 4 \sec^{2} (x)}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.5.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$lim_{x \to \frac{π}{4}} \frac{- 4 {(\frac{1}{\cos (x)})}^{2}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.5.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

$lim_{x \to \frac{π}{4}} \frac{- 4 \frac{1^{2}}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.5.4

One to any power is one.

$lim_{x \to \frac{π}{4}} \frac{- 4 \frac{1}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.5.5

Combine $- 4$ and $\frac{1}{\cos^{2} (x)}$ .

$lim_{x \to \frac{π}{4}} \frac{\frac{- 4}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.5.6

Move the negative in front of the fraction.

$lim_{x \to \frac{π}{4}} \frac{- \frac{4}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 3.6

By the Sum Rule, the derivative of $\sin (x) - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]$ .

$lim_{x \to \frac{π}{4}} \frac{- \frac{4}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]}$

Step 3.7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 3.8

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 3.8.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$lim_{x \to \frac{π}{4}} \frac{- \frac{4}{\cos^{2} (x)}}{\cos (x) - \frac{d}{d x} [\cos (x)]}$

Step 3.8.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to \frac{π}{4}} \frac{- \frac{4}{\cos^{2} (x)}}{\cos (x) - - \sin (x)}$

Step 3.8.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to \frac{π}{4}} \frac{- \frac{4}{\cos^{2} (x)}}{\cos (x) + 1 \sin (x)}$

Step 3.8.4

Multiply $\sin (x)$ by $1$ .

$lim_{x \to \frac{π}{4}} \frac{- \frac{4}{\cos^{2} (x)}}{\cos (x) + \sin (x)}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \frac{π}{4}} - \frac{4}{\cos^{2} (x)} \cdot \frac{1}{\cos (x) + \sin (x)}$

Step 5

Multiply $\frac{1}{\cos (x) + \sin (x)}$ by $\frac{4}{\cos^{2} (x)}$ .

$lim_{x \to \frac{π}{4}} - \frac{4}{(\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 6

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to \frac{π}{4}} \frac{4}{(\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 7

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$- 4 lim_{x \to \frac{π}{4}} \frac{1}{(\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 8

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$- 4 \frac{lim_{x \to \frac{π}{4}} 1}{lim_{x \to \frac{π}{4}} (\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{4}$ .

$- 4 \frac{1}{lim_{x \to \frac{π}{4}} (\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 10

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$- 4 \frac{1}{lim_{x \to \frac{π}{4}} \cos (x) + \sin (x) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 11

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$- 4 \frac{1}{(lim_{x \to \frac{π}{4}} \cos (x) + lim_{x \to \frac{π}{4}} \sin (x)) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 12

Move the limit inside the trig function because cosine is continuous.

$- 4 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + lim_{x \to \frac{π}{4}} \sin (x)) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 13

Move the limit inside the trig function because sine is continuous.

$- 4 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 14

Move the exponent $2$ from $\cos^{2} (x)$ outside the limit using the Limits Power Rule.

$- 4 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot {(lim_{x \to \frac{π}{4}} \cos (x))}^{2}}$

Step 15

Move the limit inside the trig function because cosine is continuous.

$- 4 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot \cos^{2} (lim_{x \to \frac{π}{4}} x)}$

Step 16

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 16.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$- 4 \frac{1}{(\cos (\frac{π}{4}) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot \cos^{2} (lim_{x \to \frac{π}{4}} x)}$

Step 16.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$- 4 \frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot \cos^{2} (lim_{x \to \frac{π}{4}} x)}$

Step 16.3

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$- 4 \frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot \cos^{2} (\frac{π}{4})}$

Step 17

Simplify the answer.

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Step 17.1

Multiply by $1$ .

$- 4 \frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot (\cos^{2} (\frac{π}{4}) \cdot 1)}$

Step 17.2

Separate fractions.

$- 4 (\frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot (1)} \cdot \frac{1}{\cos^{2} (\frac{π}{4})})$

Step 17.3

Convert from $\frac{1}{\cos^{2} (\frac{π}{4})}$ to $\sec^{2} (\frac{π}{4})$ .

$- 4 (\frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot (1)} \sec^{2} (\frac{π}{4}))$

Step 17.4

Multiply $\cos (\frac{π}{4}) + \sin (\frac{π}{4})$ by $1$ .

$- 4 (\frac{1}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})} \sec^{2} (\frac{π}{4}))$

Step 17.5

Simplify the denominator.

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Step 17.5.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 4 (\frac{1}{\frac{\sqrt{2}}{2} + \sin (\frac{π}{4})} \sec^{2} (\frac{π}{4}))$

Step 17.5.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 17.5.3

Combine the numerators over the common denominator.

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

Step 17.5.4

Rewrite $\frac{\sqrt{2} + \sqrt{2}}{2}$ in a factored form.

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Step 17.5.4.1

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

$- 4 (\frac{1}{\frac{2 \sqrt{2}}{2}} \sec^{2} (\frac{π}{4}))$

Step 17.5.4.2

Reduce the expression by cancelling the common factors.

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Step 17.5.4.2.1

Reduce the expression $\frac{2 \sqrt{2}}{2}$ by cancelling the common factors.

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Step 17.5.4.2.1.1

Cancel the common factor.

$- 4 (\frac{1}{\frac{2 \sqrt{2}}{2}} \sec^{2} (\frac{π}{4}))$

Step 17.5.4.2.1.2

Rewrite the expression.

$- 4 (\frac{1}{\frac{\sqrt{2}}{1}} \sec^{2} (\frac{π}{4}))$

Step 17.5.4.2.2

Divide $\sqrt{2}$ by $1$ .

$- 4 (\frac{1}{\sqrt{2}} \sec^{2} (\frac{π}{4}))$

Step 17.6

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

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

Step 17.7

Combine and simplify the denominator.

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Step 17.7.1

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

$- 4 (\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} \sec^{2} (\frac{π}{4}))$

Step 17.7.2

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

$- 4 (\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \sec^{2} (\frac{π}{4}))$

Step 17.7.3

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

$- 4 (\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \sec^{2} (\frac{π}{4}))$

Step 17.7.4

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

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

Step 17.7.5

Add $1$ and $1$ .

$- 4 (\frac{\sqrt{2}}{{\sqrt{2}}^{2}} \sec^{2} (\frac{π}{4}))$

Step 17.7.6

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

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Step 17.7.6.1

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

$- 4 (\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \sec^{2} (\frac{π}{4}))$

Step 17.7.6.2

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

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

Step 17.7.6.3

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

$- 4 (\frac{\sqrt{2}}{2^{\frac{2}{2}}} \sec^{2} (\frac{π}{4}))$

Step 17.7.6.4

Cancel the common factor of $2$ .

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Step 17.7.6.4.1

Cancel the common factor.

$- 4 (\frac{\sqrt{2}}{2^{\frac{2}{2}}} \sec^{2} (\frac{π}{4}))$

Step 17.7.6.4.2

Rewrite the expression.

$- 4 (\frac{\sqrt{2}}{2^{1}} \sec^{2} (\frac{π}{4}))$

Step 17.7.6.5

Evaluate the exponent.

$- 4 (\frac{\sqrt{2}}{2} \sec^{2} (\frac{π}{4}))$

Step 17.8

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

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

Step 17.9

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

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

Step 17.10

Combine and simplify the denominator.

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Step 17.10.1

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

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

Step 17.10.2

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

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

Step 17.10.3

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

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

Step 17.10.4

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

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

Step 17.10.5

Add $1$ and $1$ .

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

Step 17.10.6

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

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Step 17.10.6.1

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

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

Step 17.10.6.2

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

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

Step 17.10.6.3

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

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

Step 17.10.6.4

Cancel the common factor of $2$ .

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Step 17.10.6.4.1

Cancel the common factor.

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

Step 17.10.6.4.2

Rewrite the expression.

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

Step 17.10.6.5

Evaluate the exponent.

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

Step 17.11

Cancel the common factor of $2$ .

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Step 17.11.1

Cancel the common factor.

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

Step 17.11.2

Divide $\sqrt{2}$ by $1$ .

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

Step 17.12

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

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Step 17.12.1

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

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

Step 17.12.2

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

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

Step 17.12.3

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

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

Step 17.12.4

Cancel the common factor of $2$ .

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Step 17.12.4.1

Cancel the common factor.

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

Step 17.12.4.2

Rewrite the expression.

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

Step 17.12.5

Evaluate the exponent.

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

Step 17.13

Cancel the common factor of $2$ .

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Step 17.13.1

Cancel the common factor.

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

Step 17.13.2

Rewrite the expression.

$- 4 \sqrt{2}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$- 4 \sqrt{2}$

Decimal Form:

$- 5.65685424 \dots$