Calculus Examples

$lim_{x \to \frac{π}{4}} \frac{7 - 7 \tan (x)}{\sin (x) - \cos (x)}$

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{lim_{x \to \frac{π}{4}} 7 - 7 \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.1.2.1

Evaluate the limit.

Tap for more steps...

Step 1.1.2.1.1

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

$\frac{lim_{x \to \frac{π}{4}} 7 - lim_{x \to \frac{π}{4}} 7 \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.1.2

Evaluate the limit of $7$ which is constant as $x$ approaches $\frac{π}{4}$ .

$\frac{7 - lim_{x \to \frac{π}{4}} 7 \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.1.3

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 - 7 lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.1.4

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

$\frac{7 - 7 \tan (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.2

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

$\frac{7 - 7 \tan (\frac{π}{4})}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.3

Simplify the answer.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

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

$\frac{7 - 7 \cdot 1}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.3.1.2

Multiply $- 7$ by $1$ .

$\frac{7 - 7}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.3.2

Subtract $7$ from $7$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.1.3.1

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 1.1.3.2

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 1.1.3.3

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 1.1.3.4

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

Tap for more steps...

Step 1.1.3.4.1

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 1.1.3.4.2

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

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

Step 1.1.3.5

Simplify the answer.

Tap for more steps...

Step 1.1.3.5.1

Simplify each term.

Tap for more steps...

Step 1.1.3.5.1.1

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

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

Step 1.1.3.5.1.2

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

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

Step 1.1.3.5.2

Combine the numerators over the common denominator.

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

Step 1.1.3.5.3

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

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

Step 1.1.3.5.4

Divide $0$ by $2$ .

$\frac{0}{0}$

Step 1.1.3.5.5

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to \frac{π}{4}} \frac{7 - 7 \tan (x)}{\sin (x) - \cos (x)} = lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [7 - 7 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [7 - 7 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.2

By the Sum Rule, the derivative of $7 - 7 \tan (x)$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 7 \tan (x)]$ .

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [7] + \frac{d}{d x} [- 7 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$lim_{x \to \frac{π}{4}} \frac{0 + \frac{d}{d x} [- 7 \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 7 \tan (x)]$ .

Tap for more steps...

Step 1.3.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 \tan (x)$ with respect to $x$ is $- 7 \frac{d}{d x} [\tan (x)]$ .

$lim_{x \to \frac{π}{4}} \frac{0 - 7 \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.4.2

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

$lim_{x \to \frac{π}{4}} \frac{0 - 7 \sec^{2} (x)}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.5

Simplify.

Tap for more steps...

Step 1.3.5.1

Subtract $7 \sec^{2} (x)$ from $0$ .

$lim_{x \to \frac{π}{4}} \frac{- 7 \sec^{2} (x)}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.5.2

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

$lim_{x \to \frac{π}{4}} \frac{- 7 {(\frac{1}{\cos (x)})}^{2}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.5.3

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

$lim_{x \to \frac{π}{4}} \frac{- 7 \frac{1^{2}}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.5.4

One to any power is one.

$lim_{x \to \frac{π}{4}} \frac{- 7 \frac{1}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.5.5

Combine $- 7$ and $\frac{1}{\cos^{2} (x)}$ .

$lim_{x \to \frac{π}{4}} \frac{\frac{- 7}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.5.6

Move the negative in front of the fraction.

$lim_{x \to \frac{π}{4}} \frac{- \frac{7}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.6

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

$lim_{x \to \frac{π}{4}} \frac{- \frac{7}{\cos^{2} (x)}}{\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]}$

Step 1.3.7

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

$lim_{x \to \frac{π}{4}} \frac{- \frac{7}{\cos^{2} (x)}}{\cos (x) + \frac{d}{d x} [- \cos (x)]}$

Step 1.3.8

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

Tap for more steps...

Step 1.3.8.1

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

$lim_{x \to \frac{π}{4}} \frac{- \frac{7}{\cos^{2} (x)}}{\cos (x) - \frac{d}{d x} [\cos (x)]}$

Step 1.3.8.2

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

$lim_{x \to \frac{π}{4}} \frac{- \frac{7}{\cos^{2} (x)}}{\cos (x) - - \sin (x)}$

Step 1.3.8.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to \frac{π}{4}} \frac{- \frac{7}{\cos^{2} (x)}}{\cos (x) + 1 \sin (x)}$

Step 1.3.8.4

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

$lim_{x \to \frac{π}{4}} \frac{- \frac{7}{\cos^{2} (x)}}{\cos (x) + \sin (x)}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \frac{π}{4}} - \frac{7}{\cos^{2} (x)} \cdot \frac{1}{\cos (x) + \sin (x)}$

Step 1.5

Multiply $\frac{1}{\cos (x) + \sin (x)}$ by $\frac{7}{\cos^{2} (x)}$ .

$lim_{x \to \frac{π}{4}} - \frac{7}{(\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

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

$- lim_{x \to \frac{π}{4}} \frac{7}{(\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 2.2

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$- 7 lim_{x \to \frac{π}{4}} \frac{1}{(\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 2.3

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

$- 7 \frac{lim_{x \to \frac{π}{4}} 1}{lim_{x \to \frac{π}{4}} (\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 2.4

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

$- 7 \frac{1}{lim_{x \to \frac{π}{4}} (\cos (x) + \sin (x)) \cos^{2} (x)}$

Step 2.5

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

$- 7 \frac{1}{lim_{x \to \frac{π}{4}} \cos (x) + \sin (x) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 2.6

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

$- 7 \frac{1}{(lim_{x \to \frac{π}{4}} \cos (x) + lim_{x \to \frac{π}{4}} \sin (x)) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 2.7

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

$- 7 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + lim_{x \to \frac{π}{4}} \sin (x)) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 2.8

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

$- 7 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot lim_{x \to \frac{π}{4}} \cos^{2} (x)}$

Step 2.9

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

$- 7 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot {(lim_{x \to \frac{π}{4}} \cos (x))}^{2}}$

Step 2.10

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

$- 7 \frac{1}{(\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot \cos^{2} (lim_{x \to \frac{π}{4}} x)}$

Step 3

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

Tap for more steps...

Step 3.1

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

$- 7 \frac{1}{(\cos (\frac{π}{4}) + \sin (lim_{x \to \frac{π}{4}} x)) \cdot \cos^{2} (lim_{x \to \frac{π}{4}} x)}$

Step 3.2

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

$- 7 \frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot \cos^{2} (lim_{x \to \frac{π}{4}} x)}$

Step 3.3

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

$- 7 \frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot \cos^{2} (\frac{π}{4})}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Multiply by $1$ .

$- 7 \frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot (\cos^{2} (\frac{π}{4}) \cdot 1)}$

Step 4.2

Separate fractions.

$- 7 (\frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot (1)} \cdot \frac{1}{\cos^{2} (\frac{π}{4})})$

Step 4.3

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

$- 7 (\frac{1}{(\cos (\frac{π}{4}) + \sin (\frac{π}{4})) \cdot (1)} \sec^{2} (\frac{π}{4}))$

Step 4.4

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

$- 7 (\frac{1}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})} \sec^{2} (\frac{π}{4}))$

Step 4.5

Simplify the denominator.

Tap for more steps...

Step 4.5.1

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

$- 7 (\frac{1}{\frac{\sqrt{2}}{2} + \sin (\frac{π}{4})} \sec^{2} (\frac{π}{4}))$

Step 4.5.2

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

$- 7 (\frac{1}{\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}} \sec^{2} (\frac{π}{4}))$

Step 4.5.3

Combine the numerators over the common denominator.

$- 7 (\frac{1}{\frac{\sqrt{2} + \sqrt{2}}{2}} \sec^{2} (\frac{π}{4}))$

Step 4.5.4

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

Tap for more steps...

Step 4.5.4.1

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

$- 7 (\frac{1}{\frac{2 \sqrt{2}}{2}} \sec^{2} (\frac{π}{4}))$

Step 4.5.4.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.5.4.2.1

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

Tap for more steps...

Step 4.5.4.2.1.1

Cancel the common factor.

$- 7 (\frac{1}{\frac{2 \sqrt{2}}{2}} \sec^{2} (\frac{π}{4}))$

Step 4.5.4.2.1.2

Rewrite the expression.

$- 7 (\frac{1}{\frac{\sqrt{2}}{1}} \sec^{2} (\frac{π}{4}))$

Step 4.5.4.2.2

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

$- 7 (\frac{1}{\sqrt{2}} \sec^{2} (\frac{π}{4}))$

Step 4.6

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

$- 7 (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \sec^{2} (\frac{π}{4}))$

Step 4.7

Combine and simplify the denominator.

Tap for more steps...

Step 4.7.1

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

$- 7 (\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} \sec^{2} (\frac{π}{4}))$

Step 4.7.2

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

$- 7 (\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \sec^{2} (\frac{π}{4}))$

Step 4.7.3

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

$- 7 (\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \sec^{2} (\frac{π}{4}))$

Step 4.7.4

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

$- 7 (\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} \sec^{2} (\frac{π}{4}))$

Step 4.7.5

Add $1$ and $1$ .

$- 7 (\frac{\sqrt{2}}{{\sqrt{2}}^{2}} \sec^{2} (\frac{π}{4}))$

Step 4.7.6

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

Tap for more steps...

Step 4.7.6.1

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

$- 7 (\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \sec^{2} (\frac{π}{4}))$

Step 4.7.6.2

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

$- 7 (\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \sec^{2} (\frac{π}{4}))$

Step 4.7.6.3

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

$- 7 (\frac{\sqrt{2}}{2^{\frac{2}{2}}} \sec^{2} (\frac{π}{4}))$

Step 4.7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.7.6.4.1

Cancel the common factor.

$- 7 (\frac{\sqrt{2}}{2^{\frac{2}{2}}} \sec^{2} (\frac{π}{4}))$

Step 4.7.6.4.2

Rewrite the expression.

$- 7 (\frac{\sqrt{2}}{2^{1}} \sec^{2} (\frac{π}{4}))$

Step 4.7.6.5

Evaluate the exponent.

$- 7 (\frac{\sqrt{2}}{2} \sec^{2} (\frac{π}{4}))$

Step 4.8

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

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2}{\sqrt{2}})}^{2})$

Step 4.9

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

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2})$

Step 4.10

Combine and simplify the denominator.

Tap for more steps...

Step 4.10.1

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

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2})$

Step 4.10.2

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

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

Step 4.10.3

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

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

Step 4.10.4

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

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

Step 4.10.5

Add $1$ and $1$ .

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2})$

Step 4.10.6

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

Tap for more steps...

Step 4.10.6.1

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

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

Step 4.10.6.2

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

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

Step 4.10.6.3

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

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2})$

Step 4.10.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.10.6.4.1

Cancel the common factor.

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2})$

Step 4.10.6.4.2

Rewrite the expression.

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

Step 4.10.6.5

Evaluate the exponent.

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2 \sqrt{2}}{2})}^{2})$

Step 4.11

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.11.1

Cancel the common factor.

$- 7 (\frac{\sqrt{2}}{2} {(\frac{2 \sqrt{2}}{2})}^{2})$

Step 4.11.2

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

$- 7 (\frac{\sqrt{2}}{2} {\sqrt{2}}^{2})$

Step 4.12

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

Tap for more steps...

Step 4.12.1

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

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

Step 4.12.2

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

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

Step 4.12.3

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

$- 7 (\frac{\sqrt{2}}{2} \cdot 2^{\frac{2}{2}})$

Step 4.12.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.12.4.1

Cancel the common factor.

$- 7 (\frac{\sqrt{2}}{2} \cdot 2^{\frac{2}{2}})$

Step 4.12.4.2

Rewrite the expression.

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

Step 4.12.5

Evaluate the exponent.

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

Step 4.13

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.13.1

Cancel the common factor.

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

Step 4.13.2

Rewrite the expression.

$- 7 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- 7 \sqrt{2}$

Decimal Form:

$- 9.89949493 \dots$