Calculus Examples

$\int_{0}^{\frac{π}{4}} \sin^{5} (x) d x$

Step 1

Factor out $\sin^{4} (x)$ .

$\int_{0}^{\frac{π}{4}} \sin^{4} (x) \sin (x) d x$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $4$ .

$\int_{0}^{\frac{π}{4}} {\sin (x)}^{2 (2)} \sin (x) d x$

Step 2.2

Rewrite ${\sin (x)}^{2 (2)}$ as exponentiation.

$\int_{0}^{\frac{π}{4}} {(\sin^{2} (x))}^{2} \sin (x) d x$

Step 3

Using the Pythagorean Identity, rewrite $\sin^{2} (x)$ as $1 - \cos^{2} (x)$ .

$\int_{0}^{\frac{π}{4}} {(1 - \cos^{2} (x))}^{2} \sin (x) d x$

Step 4

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- \frac{1}{\sin (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $\cos (x)$ .

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

Step 4.1.2

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

$- \sin (x)$

Step 4.2

Substitute the lower limit in for $x$ in $u = \cos (x)$ .

$u_{lower} = \cos (0)$

Step 4.3

The exact value of $\cos (0)$ is $1$ .

$u_{lower} = 1$

Step 4.4

Substitute the upper limit in for $x$ in $u = \cos (x)$ .

$u_{upper} = \cos (\frac{π}{4})$

Step 4.5

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

$u_{upper} = \frac{\sqrt{2}}{2}$

Step 4.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = \frac{\sqrt{2}}{2}$

Step 4.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{\frac{\sqrt{2}}{2}} - {(1 - u^{2})}^{2} d u$

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{1}^{\frac{\sqrt{2}}{2}} {(1 - u^{2})}^{2} d u$

Step 6

Expand ${(1 - u^{2})}^{2}$ .

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Step 6.1

Rewrite ${(1 - u^{2})}^{2}$ as $(1 - u^{2}) (1 - u^{2})$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} (1 - u^{2}) (1 - u^{2}) d u$

Step 6.2

Apply the distributive property.

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 (1 - u^{2}) - u^{2} (1 - u^{2}) d u$

Step 6.3

Apply the distributive property.

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 \cdot 1 + 1 (- u^{2}) - u^{2} (1 - u^{2}) d u$

Step 6.4

Apply the distributive property.

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 \cdot 1 + 1 (- u^{2}) - u^{2} \cdot 1 - u^{2} (- u^{2}) d u$

Step 6.5

Move $u^{2}$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 \cdot 1 + 1 \cdot - 1 u^{2} - 1 \cdot 1 u^{2} - u^{2} (- u^{2}) d u$

Step 6.6

Move $u^{2}$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 \cdot 1 + 1 \cdot - 1 u^{2} - 1 \cdot 1 u^{2} - 1 \cdot - 1 u^{2} u^{2} d u$

Step 6.7

Multiply $1$ by $1$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 + 1 \cdot - 1 u^{2} - 1 \cdot 1 u^{2} - 1 \cdot - 1 u^{2} u^{2} d u$

Step 6.8

Multiply $- 1$ by $1$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 - u^{2} - 1 \cdot 1 u^{2} - 1 \cdot - 1 u^{2} u^{2} d u$

Step 6.9

Multiply $- 1$ by $1$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 - u^{2} - u^{2} - 1 \cdot - 1 u^{2} u^{2} d u$

Step 6.10

Multiply $- 1$ by $- 1$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 - u^{2} - u^{2} + 1 u^{2} u^{2} d u$

Step 6.11

Multiply $u^{2}$ by $1$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 - u^{2} - u^{2} + u^{2} u^{2} d u$

Step 6.12

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

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 - u^{2} - u^{2} + u^{2 + 2} d u$

Step 6.13

Add $2$ and $2$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 - u^{2} - u^{2} + u^{4} d u$

Step 6.14

Subtract $u^{2}$ from $- u^{2}$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 - 2 u^{2} + u^{4} d u$

Step 6.15

Reorder $- 2 u^{2}$ and $u^{4}$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} 1 + u^{4} - 2 u^{2} d u$

Step 6.16

Move $1$ .

$- \int_{1}^{\frac{\sqrt{2}}{2}} u^{4} - 2 u^{2} + 1 d u$

Step 7

Split the single integral into multiple integrals.

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

Step 8

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$- (\frac{1}{5} u^{5}]_{1}^{\frac{\sqrt{2}}{2}} + \int_{1}^{\frac{\sqrt{2}}{2}} - 2 u^{2} d u + \int_{1}^{\frac{\sqrt{2}}{2}} d u)$

Step 9

Combine $\frac{1}{5}$ and $u^{5}$ .

$- (\frac{u^{5}}{5}]_{1}^{\frac{\sqrt{2}}{2}} + \int_{1}^{\frac{\sqrt{2}}{2}} - 2 u^{2} d u + \int_{1}^{\frac{\sqrt{2}}{2}} d u)$

Step 10

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$- (\frac{u^{5}}{5}]_{1}^{\frac{\sqrt{2}}{2}} - 2 \int_{1}^{\frac{\sqrt{2}}{2}} u^{2} d u + \int_{1}^{\frac{\sqrt{2}}{2}} d u)$

Step 11

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$- (\frac{u^{5}}{5}]_{1}^{\frac{\sqrt{2}}{2}} - 2 (\frac{1}{3} u^{3}]_{1}^{\frac{\sqrt{2}}{2}}) + \int_{1}^{\frac{\sqrt{2}}{2}} d u)$

Step 12

Combine $\frac{1}{3}$ and $u^{3}$ .

$- (\frac{u^{5}}{5}]_{1}^{\frac{\sqrt{2}}{2}} - 2 (\frac{u^{3}}{3}]_{1}^{\frac{\sqrt{2}}{2}}) + \int_{1}^{\frac{\sqrt{2}}{2}} d u)$

Step 13

Apply the constant rule.

$- (\frac{u^{5}}{5}]_{1}^{\frac{\sqrt{2}}{2}} - 2 (\frac{u^{3}}{3}]_{1}^{\frac{\sqrt{2}}{2}}) + u]_{1}^{\frac{\sqrt{2}}{2}})$

Step 14

Combine $\frac{u^{5}}{5}]_{1}^{\frac{\sqrt{2}}{2}}$ and $u]_{1}^{\frac{\sqrt{2}}{2}}$ .

$- (\frac{u^{5}}{5} + u]_{1}^{\frac{\sqrt{2}}{2}} - 2 (\frac{u^{3}}{3}]_{1}^{\frac{\sqrt{2}}{2}}))$

Step 15

Substitute and simplify.

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Step 15.1

Evaluate $\frac{u^{5}}{5} + u$ at $\frac{\sqrt{2}}{2}$ and at $1$ .

$- ((\frac{{(\frac{\sqrt{2}}{2})}^{5}}{5} + \frac{\sqrt{2}}{2}) - (\frac{1^{5}}{5} + 1) - 2 (\frac{u^{3}}{3}]_{1}^{\frac{\sqrt{2}}{2}}))$

Step 15.2

Evaluate $\frac{u^{3}}{3}$ at $\frac{\sqrt{2}}{2}$ and at $1$ .

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

Step 15.3

Simplify.

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Step 15.3.1

One to any power is one.

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

Step 15.3.2

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

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

Step 15.3.3

Combine the numerators over the common denominator.

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

Step 15.3.4

Add $1$ and $5$ .

$- (\frac{{(\frac{\sqrt{2}}{2})}^{5}}{5} + \frac{\sqrt{2}}{2} - \frac{6}{5} - 2 ((\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.5

One to any power is one.

$- (\frac{{(\frac{\sqrt{2}}{2})}^{5}}{5} + \frac{\sqrt{2}}{2} - \frac{6}{5} - 2 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}))$

Step 15.3.6

To write $- 2 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3})$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 15.3.7

Combine $- 2 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3})$ and $\frac{5}{5}$ .

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

Step 15.3.8

Combine the numerators over the common denominator.

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

Step 15.3.9

Multiply $5$ by $- 2$ .

$- (\frac{{(\frac{\sqrt{2}}{2})}^{5} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3})}{5} + \frac{\sqrt{2}}{2} - \frac{6}{5})$

Step 16

Simplify.

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Step 16.1

Combine the numerators over the common denominator.

$- (\frac{{(\frac{\sqrt{2}}{2})}^{5} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2

Simplify each term.

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Step 16.2.1

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

$- (\frac{\frac{{\sqrt{2}}^{5}}{2^{5}} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.2

Simplify the numerator.

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Step 16.2.2.1

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$- (\frac{\frac{\sqrt{2^{5}}}{2^{5}} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.2.2

Raise $2$ to the power of $5$ .

$- (\frac{\frac{\sqrt{32}}{2^{5}} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.2.3

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

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Step 16.2.2.3.1

Factor $16$ out of $32$ .

$- (\frac{\frac{\sqrt{16 (2)}}{2^{5}} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.2.3.2

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

$- (\frac{\frac{\sqrt{4^{2} \cdot 2}}{2^{5}} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.2.4

Pull terms out from under the radical.

$- (\frac{\frac{4 \sqrt{2}}{2^{5}} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.3

Raise $2$ to the power of $5$ .

$- (\frac{\frac{4 \sqrt{2}}{32} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.4

Cancel the common factor of $4$ and $32$ .

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Step 16.2.4.1

Factor $4$ out of $4 \sqrt{2}$ .

$- (\frac{\frac{4 (\sqrt{2})}{32} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.4.2

Cancel the common factors.

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Step 16.2.4.2.1

Factor $4$ out of $32$ .

$- (\frac{\frac{4 \sqrt{2}}{4 \cdot 8} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.4.2.2

Cancel the common factor.

$- (\frac{\frac{4 \sqrt{2}}{4 \cdot 8} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.4.2.3

Rewrite the expression.

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{{(\frac{\sqrt{2}}{2})}^{3}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5

Simplify each term.

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Step 16.2.5.1

Simplify the numerator.

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Step 16.2.5.1.1

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

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{{\sqrt{2}}^{3}}{2^{3}}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.2

Simplify the numerator.

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Step 16.2.5.1.2.1

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

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{\sqrt{2^{3}}}{2^{3}}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.2.2

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

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{\sqrt{8}}{2^{3}}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.2.3

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

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Step 16.2.5.1.2.3.1

Factor $4$ out of $8$ .

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{\sqrt{4 (2)}}{2^{3}}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.2.3.2

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

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{\sqrt{2^{2} \cdot 2}}{2^{3}}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.2.4

Pull terms out from under the radical.

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{2 \sqrt{2}}{2^{3}}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.3

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

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{2 \sqrt{2}}{8}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.4

Cancel the common factor of $2$ and $8$ .

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Step 16.2.5.1.4.1

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

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{2 (\sqrt{2})}{8}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.4.2

Cancel the common factors.

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Step 16.2.5.1.4.2.1

Factor $2$ out of $8$ .

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{2 \sqrt{2}}{2 \cdot 4}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.4.2.2

Cancel the common factor.

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{2 \sqrt{2}}{2 \cdot 4}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.1.4.2.3

Rewrite the expression.

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\frac{\sqrt{2}}{4}}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.2

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\sqrt{2}}{4} \cdot \frac{1}{3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.3

Multiply $\frac{\sqrt{2}}{4} \cdot \frac{1}{3}$ .

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Step 16.2.5.3.1

Multiply $\frac{\sqrt{2}}{4}$ by $\frac{1}{3}$ .

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\sqrt{2}}{4 \cdot 3} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.5.3.2

Multiply $4$ by $3$ .

$- (\frac{\frac{\sqrt{2}}{8} - 10 (\frac{\sqrt{2}}{12} - \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.6

Apply the distributive property.

$- (\frac{\frac{\sqrt{2}}{8} - 10 \frac{\sqrt{2}}{12} - 10 (- \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.7

Cancel the common factor of $2$ .

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Step 16.2.7.1

Factor $2$ out of $- 10$ .

$- (\frac{\frac{\sqrt{2}}{8} + 2 (- 5) \frac{\sqrt{2}}{12} - 10 (- \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.7.2

Factor $2$ out of $12$ .

$- (\frac{\frac{\sqrt{2}}{8} + 2 \cdot - 5 \frac{\sqrt{2}}{2 \cdot 6} - 10 (- \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.7.3

Cancel the common factor.

$- (\frac{\frac{\sqrt{2}}{8} + 2 \cdot - 5 \frac{\sqrt{2}}{2 \cdot 6} - 10 (- \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.7.4

Rewrite the expression.

$- (\frac{\frac{\sqrt{2}}{8} - 5 \frac{\sqrt{2}}{6} - 10 (- \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.8

Combine $- 5$ and $\frac{\sqrt{2}}{6}$ .

$- (\frac{\frac{\sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{6} - 10 (- \frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.9

Multiply $- 10 (- \frac{1}{3})$ .

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Step 16.2.9.1

Multiply $- 1$ by $- 10$ .

$- (\frac{\frac{\sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{6} + 10 (\frac{1}{3}) - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.9.2

Combine $10$ and $\frac{1}{3}$ .

$- (\frac{\frac{\sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{6} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.2.10

Move the negative in front of the fraction.

$- (\frac{\frac{\sqrt{2}}{8} - \frac{5 \sqrt{2}}{6} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.3

To write $\frac{\sqrt{2}}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- (\frac{\frac{\sqrt{2}}{8} \cdot \frac{3}{3} - \frac{5 \sqrt{2}}{6} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.4

To write $- \frac{5 \sqrt{2}}{6}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- (\frac{\frac{\sqrt{2}}{8} \cdot \frac{3}{3} - \frac{5 \sqrt{2}}{6} \cdot \frac{4}{4} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.5

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.5.1

Multiply $\frac{\sqrt{2}}{8}$ by $\frac{3}{3}$ .

$- (\frac{\frac{\sqrt{2} \cdot 3}{8 \cdot 3} - \frac{5 \sqrt{2}}{6} \cdot \frac{4}{4} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.5.2

Multiply $8$ by $3$ .

$- (\frac{\frac{\sqrt{2} \cdot 3}{24} - \frac{5 \sqrt{2}}{6} \cdot \frac{4}{4} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.5.3

Multiply $\frac{5 \sqrt{2}}{6}$ by $\frac{4}{4}$ .

$- (\frac{\frac{\sqrt{2} \cdot 3}{24} - \frac{5 \sqrt{2} \cdot 4}{6 \cdot 4} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.5.4

Multiply $6$ by $4$ .

$- (\frac{\frac{\sqrt{2} \cdot 3}{24} - \frac{5 \sqrt{2} \cdot 4}{24} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.6

Combine the numerators over the common denominator.

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4}{24} + \frac{10}{3} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.7

Find the common denominator.

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Step 16.7.1

Multiply $\frac{10}{3}$ by $\frac{8}{8}$ .

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4}{24} + \frac{10}{3} \cdot \frac{8}{8} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.7.2

Multiply $\frac{10}{3}$ by $\frac{8}{8}$ .

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4}{24} + \frac{10 \cdot 8}{3 \cdot 8} - 6}{5} + \frac{\sqrt{2}}{2})$

Step 16.7.3

Write $- 6$ as a fraction with denominator $1$ .

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4}{24} + \frac{10 \cdot 8}{3 \cdot 8} + \frac{- 6}{1}}{5} + \frac{\sqrt{2}}{2})$

Step 16.7.4

Multiply $\frac{- 6}{1}$ by $\frac{24}{24}$ .

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4}{24} + \frac{10 \cdot 8}{3 \cdot 8} + \frac{- 6}{1} \cdot \frac{24}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.7.5

Multiply $\frac{- 6}{1}$ by $\frac{24}{24}$ .

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4}{24} + \frac{10 \cdot 8}{3 \cdot 8} + \frac{- 6 \cdot 24}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.7.6

Multiply $3$ by $8$ .

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4}{24} + \frac{10 \cdot 8}{24} + \frac{- 6 \cdot 24}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.8

Combine the numerators over the common denominator.

$- (\frac{\frac{\sqrt{2} \cdot 3 - 5 \sqrt{2} \cdot 4 + 10 \cdot 8 - 6 \cdot 24}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.9

Simplify each term.

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Step 16.9.1

Move $3$ to the left of $\sqrt{2}$ .

$- (\frac{\frac{3 \cdot \sqrt{2} - 5 \sqrt{2} \cdot 4 + 10 \cdot 8 - 6 \cdot 24}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.9.2

Multiply $4$ by $- 5$ .

$- (\frac{\frac{3 \sqrt{2} - 20 \sqrt{2} + 10 \cdot 8 - 6 \cdot 24}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.9.3

Multiply $10$ by $8$ .

$- (\frac{\frac{3 \sqrt{2} - 20 \sqrt{2} + 80 - 6 \cdot 24}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.9.4

Multiply $- 6$ by $24$ .

$- (\frac{\frac{3 \sqrt{2} - 20 \sqrt{2} + 80 - 144}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.10

Subtract $20 \sqrt{2}$ from $3 \sqrt{2}$ .

$- (\frac{\frac{- 17 \sqrt{2} + 80 - 144}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.11

Subtract $144$ from $80$ .

$- (\frac{\frac{- 17 \sqrt{2} - 64}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.12

Factor $- 1$ out of $- 17 \sqrt{2}$ .

$- (\frac{\frac{- (17 \sqrt{2}) - 64}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.13

Rewrite $- 64$ as $- 1 (64)$ .

$- (\frac{\frac{- (17 \sqrt{2}) - 1 (64)}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.14

Factor $- 1$ out of $- (17 \sqrt{2}) - 1 (64)$ .

$- (\frac{\frac{- (17 \sqrt{2} + 64)}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.15

Rewrite $- (17 \sqrt{2} + 64)$ as $- 1 (17 \sqrt{2} + 64)$ .

$- (\frac{\frac{- 1 (17 \sqrt{2} + 64)}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.16

Move the negative in front of the fraction.

$- (\frac{- \frac{17 \sqrt{2} + 64}{24}}{5} + \frac{\sqrt{2}}{2})$

Step 16.17

Simplify each term.

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Step 16.17.1

Multiply the numerator by the reciprocal of the denominator.

$- (- \frac{17 \sqrt{2} + 64}{24} \cdot \frac{1}{5} + \frac{\sqrt{2}}{2})$

Step 16.17.2

Multiply $- \frac{17 \sqrt{2} + 64}{24} \cdot \frac{1}{5}$ .

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Step 16.17.2.1

Multiply $\frac{1}{5}$ by $\frac{17 \sqrt{2} + 64}{24}$ .

$- (- \frac{17 \sqrt{2} + 64}{5 \cdot 24} + \frac{\sqrt{2}}{2})$

Step 16.17.2.2

Multiply $5$ by $24$ .

$- (- \frac{17 \sqrt{2} + 64}{120} + \frac{\sqrt{2}}{2})$

Step 16.18

To write $\frac{\sqrt{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{60}{60}$ .

$- (- \frac{17 \sqrt{2} + 64}{120} + \frac{\sqrt{2}}{2} \cdot \frac{60}{60})$

Step 16.19

Write each expression with a common denominator of $120$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.19.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{60}{60}$ .

$- (- \frac{17 \sqrt{2} + 64}{120} + \frac{\sqrt{2} \cdot 60}{2 \cdot 60})$

Step 16.19.2

Multiply $2$ by $60$ .

$- (- \frac{17 \sqrt{2} + 64}{120} + \frac{\sqrt{2} \cdot 60}{120})$

Step 16.20

Combine the numerators over the common denominator.

$- \frac{- (17 \sqrt{2} + 64) + \sqrt{2} \cdot 60}{120}$

Step 16.21

Simplify the numerator.

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Step 16.21.1

Apply the distributive property.

$- \frac{- (17 \sqrt{2}) - 1 \cdot 64 + \sqrt{2} \cdot 60}{120}$

Step 16.21.2

Multiply $17$ by $- 1$ .

$- \frac{- 17 \sqrt{2} - 1 \cdot 64 + \sqrt{2} \cdot 60}{120}$

Step 16.21.3

Multiply $- 1$ by $64$ .

$- \frac{- 17 \sqrt{2} - 64 + \sqrt{2} \cdot 60}{120}$

Step 16.21.4

Move $60$ to the left of $\sqrt{2}$ .

$- \frac{- 17 \sqrt{2} - 64 + 60 \cdot \sqrt{2}}{120}$

Step 16.21.5

Add $- 17 \sqrt{2}$ and $60 \sqrt{2}$ .

$- \frac{- 64 + 43 \sqrt{2}}{120}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$- \frac{- 64 + 43 \sqrt{2}}{120}$

Decimal Form:

$0.02657347 \dots$