Calculus Examples

$\int_{0}^{\frac{π}{2}} \cos^{7} (x) \sin^{5} (x) d x$

Step 1

Factor out $\cos^{6} (x)$ .

$\int_{0}^{\frac{π}{2}} \cos^{6} (x) \cos (x) \sin^{5} (x) d x$

Step 2

Simplify with factoring out.

Tap for more steps...

Step 2.1

Factor $2$ out of $6$ .

$\int_{0}^{\frac{π}{2}} {\cos (x)}^{2 (3)} \cos (x) \sin^{5} (x) d x$

Step 2.2

Rewrite ${\cos (x)}^{2 (3)}$ as exponentiation.

$\int_{0}^{\frac{π}{2}} {(\cos^{2} (x))}^{3} \cos (x) \sin^{5} (x) d x$

Step 3

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

$\int_{0}^{\frac{π}{2}} {(1 - \sin^{2} (x))}^{3} \cos (x) \sin^{5} (x) d x$

Step 4

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

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Differentiate $\sin (x)$ .

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

Step 4.1.2

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

$\cos (x)$

Step 4.2

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

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

Step 4.3

The exact value of $\sin (0)$ is $0$ .

$u_{lower} = 0$

Step 4.4

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

$u_{upper} = \sin (\frac{π}{2})$

Step 4.5

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$u_{upper} = 1$

Step 4.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 4.7

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

$\int_{0}^{1} {(1 - u^{2})}^{3} u^{5} d u$

Step 5

Expand ${(1 - u^{2})}^{3} u^{5}$ .

Tap for more steps...

Step 5.1

Use the Binomial Theorem.

$\int_{0}^{1} (1^{3} + 3 \cdot 1^{2} (- u^{2}) + 3 \cdot 1 {(- u^{2})}^{2} + {(- u^{2})}^{3}) u^{5} d u$

Step 5.2

Rewrite the exponentiation as a product.

$\int_{0}^{1} (1 \cdot 1^{2} + 3 \cdot 1^{2} (- u^{2}) + 3 \cdot 1 {(- u^{2})}^{2} + {(- u^{2})}^{3}) u^{5} d u$

Step 5.3

Rewrite the exponentiation as a product.

$\int_{0}^{1} (1 \cdot (1 \cdot 1) + 3 \cdot 1^{2} (- u^{2}) + 3 \cdot 1 {(- u^{2})}^{2} + {(- u^{2})}^{3}) u^{5} d u$

Step 5.4

Rewrite the exponentiation as a product.

$\int_{0}^{1} (1 \cdot (1 \cdot 1) + 3 \cdot (1 \cdot 1) (- u^{2}) + 3 \cdot 1 {(- u^{2})}^{2} + {(- u^{2})}^{3}) u^{5} d u$

Step 5.5

Rewrite the exponentiation as a product.

$\int_{0}^{1} (1 \cdot (1 \cdot 1) + 3 \cdot (1 \cdot 1) (- u^{2}) + 3 \cdot 1 (- u^{2} (- u^{2})) + {(- u^{2})}^{3}) u^{5} d u$

Step 5.6

Rewrite the exponentiation as a product.

$\int_{0}^{1} (1 \cdot (1 \cdot 1) + 3 \cdot (1 \cdot 1) (- u^{2}) + 3 \cdot 1 (- u^{2} (- u^{2})) - u^{2} {(- u^{2})}^{2}) u^{5} d u$

Step 5.7

Rewrite the exponentiation as a product.

$\int_{0}^{1} (1 \cdot (1 \cdot 1) + 3 \cdot (1 \cdot 1) (- u^{2}) + 3 \cdot 1 (- u^{2} (- u^{2})) - u^{2} (- u^{2} (- u^{2}))) u^{5} d u$

Step 5.8

Apply the distributive property.

$\int_{0}^{1} (1 \cdot (1 \cdot 1) + 3 \cdot (1 \cdot 1) (- u^{2}) + 3 \cdot 1 (- u^{2} (- u^{2}))) u^{5} - u^{2} (- u^{2} (- u^{2})) u^{5} d u$

Step 5.9

Apply the distributive property.

$\int_{0}^{1} (1 \cdot (1 \cdot 1) + 3 \cdot (1 \cdot 1) (- u^{2})) u^{5} + 3 \cdot 1 (- u^{2} (- u^{2})) u^{5} - u^{2} (- u^{2} (- u^{2})) u^{5} d u$

Step 5.10

Apply the distributive property.

$\int_{0}^{1} 1 \cdot (1 \cdot 1) u^{5} + 3 \cdot (1 \cdot 1) (- u^{2}) u^{5} + 3 \cdot 1 (- u^{2} (- u^{2})) u^{5} - u^{2} (- u^{2} (- u^{2})) u^{5} d u$

Step 5.11

Move $u^{2}$ .

$\int_{0}^{1} 1 \cdot 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 (- 1 \cdot - 1 u^{2} u^{2}) u^{5} - u^{2} (- u^{2} (- u^{2})) u^{5} d u$

Step 5.12

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 (- 1 \cdot - 1 u^{2}) u^{2} u^{5} - u^{2} (- u^{2} (- u^{2})) u^{5} d u$

Step 5.13

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 (- 1 \cdot - 1) u^{2} u^{2} u^{5} - u^{2} (- u^{2} (- u^{2})) u^{5} d u$

Step 5.14

Move $u^{2}$ .

$\int_{0}^{1} 1 \cdot 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - u^{2} (- 1 \cdot - 1 u^{2} u^{2}) u^{5} d u$

Step 5.15

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - u^{2} (- 1 \cdot - 1 u^{2}) u^{2} u^{5} d u$

Step 5.16

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - u^{2} (- 1 \cdot - 1) u^{2} u^{2} u^{5} d u$

Step 5.17

Move $u^{2}$ .

$\int_{0}^{1} 1 \cdot 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.18

Multiply $1$ by $1$ .

$\int_{0}^{1} 1 \cdot 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.19

Multiply $1$ by $1$ .

$\int_{0}^{1} 1 u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.20

Multiply $u^{5}$ by $1$ .

$\int_{0}^{1} u^{5} + 3 \cdot 1 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.21

Multiply $3$ by $1$ .

$\int_{0}^{1} u^{5} + 3 \cdot 1 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.22

Multiply $3$ by $1$ .

$\int_{0}^{1} u^{5} + 3 \cdot - 1 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.23

Multiply $3$ by $- 1$ .

$\int_{0}^{1} u^{5} - 3 u^{2} u^{5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.24

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

$\int_{0}^{1} u^{5} - 3 u^{2 + 5} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.25

Add $2$ and $5$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 \cdot 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.26

Multiply $3$ by $1$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.27

Multiply $3$ by $- 1$ .

$\int_{0}^{1} u^{5} - 3 u^{7} - 3 \cdot - 1 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.28

Multiply $- 3$ by $- 1$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{2} u^{2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.29

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

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{2 + 2} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.30

Add $2$ and $2$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{4} u^{5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.31

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

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{4 + 5} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.32

Add $4$ and $5$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.33

Multiply $- 1$ by $- 1$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} + 1 \cdot - 1 u^{2} u^{2} u^{2} u^{5} d u$

Step 5.34

Multiply $- 1$ by $1$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - u^{2} u^{2} u^{2} u^{5} d u$

Step 5.35

Factor out negative.

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - (u^{2} u^{2}) u^{2} u^{5} d u$

Step 5.36

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

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - u^{2 + 2} u^{2} u^{5} d u$

Step 5.37

Add $2$ and $2$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - u^{4} u^{2} u^{5} d u$

Step 5.38

Factor out negative.

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - (u^{4} u^{2}) u^{5} d u$

Step 5.39

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

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - u^{4 + 2} u^{5} d u$

Step 5.40

Add $4$ and $2$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - u^{6} u^{5} d u$

Step 5.41

Factor out negative.

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - (u^{6} u^{5}) d u$

Step 5.42

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

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - u^{6 + 5} d u$

Step 5.43

Add $6$ and $5$ .

$\int_{0}^{1} u^{5} - 3 u^{7} + 3 u^{9} - u^{11} d u$

Step 5.44

Reorder $u^{5}$ and $- 3 u^{7}$ .

$\int_{0}^{1} - 3 u^{7} + u^{5} + 3 u^{9} - u^{11} d u$

Step 5.45

Move $u^{5}$ .

$\int_{0}^{1} - 3 u^{7} + 3 u^{9} + u^{5} - u^{11} d u$

Step 5.46

Reorder $- 3 u^{7}$ and $3 u^{9}$ .

$\int_{0}^{1} 3 u^{9} - 3 u^{7} + u^{5} - u^{11} d u$

Step 5.47

Move $u^{5}$ .

$\int_{0}^{1} 3 u^{9} - 3 u^{7} - u^{11} + u^{5} d u$

Step 5.48

Move $- 3 u^{7}$ .

$\int_{0}^{1} 3 u^{9} - u^{11} - 3 u^{7} + u^{5} d u$

Step 5.49

Reorder $3 u^{9}$ and $- u^{11}$ .

$\int_{0}^{1} - u^{11} + 3 u^{9} - 3 u^{7} + u^{5} d u$

Step 6

Split the single integral into multiple integrals.

$\int_{0}^{1} - u^{11} d u + \int_{0}^{1} 3 u^{9} d u + \int_{0}^{1} - 3 u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 7

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

$- \int_{0}^{1} u^{11} d u + \int_{0}^{1} 3 u^{9} d u + \int_{0}^{1} - 3 u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 8

By the Power Rule, the integral of $u^{11}$ with respect to $u$ is $\frac{1}{12} u^{12}$ .

$- (\frac{1}{12} u^{12}]_{0}^{1}) + \int_{0}^{1} 3 u^{9} d u + \int_{0}^{1} - 3 u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 9

Combine $\frac{1}{12}$ and $u^{12}$ .

$- (\frac{u^{12}}{12}]_{0}^{1}) + \int_{0}^{1} 3 u^{9} d u + \int_{0}^{1} - 3 u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 10

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$- (\frac{u^{12}}{12}]_{0}^{1}) + 3 \int_{0}^{1} u^{9} d u + \int_{0}^{1} - 3 u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 11

By the Power Rule, the integral of $u^{9}$ with respect to $u$ is $\frac{1}{10} u^{10}$ .

$- (\frac{u^{12}}{12}]_{0}^{1}) + 3 (\frac{1}{10} u^{10}]_{0}^{1}) + \int_{0}^{1} - 3 u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 12

Combine $\frac{1}{10}$ and $u^{10}$ .

$- (\frac{u^{12}}{12}]_{0}^{1}) + 3 (\frac{u^{10}}{10}]_{0}^{1}) + \int_{0}^{1} - 3 u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 13

Since $- 3$ is constant with respect to $u$ , move $- 3$ out of the integral.

$- (\frac{u^{12}}{12}]_{0}^{1}) + 3 (\frac{u^{10}}{10}]_{0}^{1}) - 3 \int_{0}^{1} u^{7} d u + \int_{0}^{1} u^{5} d u$

Step 14

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$- (\frac{u^{12}}{12}]_{0}^{1}) + 3 (\frac{u^{10}}{10}]_{0}^{1}) - 3 (\frac{1}{8} u^{8}]_{0}^{1}) + \int_{0}^{1} u^{5} d u$

Step 15

Combine $\frac{1}{8}$ and $u^{8}$ .

$- (\frac{u^{12}}{12}]_{0}^{1}) + 3 (\frac{u^{10}}{10}]_{0}^{1}) - 3 (\frac{u^{8}}{8}]_{0}^{1}) + \int_{0}^{1} u^{5} d u$

Step 16

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$- (\frac{u^{12}}{12}]_{0}^{1}) + 3 (\frac{u^{10}}{10}]_{0}^{1}) - 3 (\frac{u^{8}}{8}]_{0}^{1}) + \frac{1}{6} u^{6}]_{0}^{1}$

Step 17

Substitute and simplify.

Tap for more steps...

Step 17.1

Evaluate $\frac{u^{12}}{12}$ at $1$ and at $0$ .

$- ((\frac{1^{12}}{12}) - \frac{0^{12}}{12}) + 3 (\frac{u^{10}}{10}]_{0}^{1}) - 3 (\frac{u^{8}}{8}]_{0}^{1}) + \frac{1}{6} u^{6}]_{0}^{1}$

Step 17.2

Evaluate $\frac{u^{10}}{10}$ at $1$ and at $0$ .

$- (\frac{1^{12}}{12} - \frac{0^{12}}{12}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{u^{8}}{8}]_{0}^{1}) + \frac{1}{6} u^{6}]_{0}^{1}$

Step 17.3

Evaluate $\frac{u^{8}}{8}$ at $1$ and at $0$ .

$- (\frac{1^{12}}{12} - \frac{0^{12}}{12}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 ((\frac{1^{8}}{8}) - \frac{0^{8}}{8}) + \frac{1}{6} u^{6}]_{0}^{1}$

Step 17.4

Evaluate $\frac{1}{6} u^{6}$ at $1$ and at $0$ .

$- (\frac{1^{12}}{12} - \frac{0^{12}}{12}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5

Simplify.

Tap for more steps...

Step 17.5.1

One to any power is one.

$- (\frac{1}{12} - \frac{0^{12}}{12}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.2

Raising $0$ to any positive power yields $0$ .

$- (\frac{1}{12} - \frac{0}{12}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.3

Cancel the common factor of $0$ and $12$ .

Tap for more steps...

Step 17.5.3.1

Factor $12$ out of $0$ .

$- (\frac{1}{12} - \frac{12 (0)}{12}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.3.2

Cancel the common factors.

Tap for more steps...

Step 17.5.3.2.1

Factor $12$ out of $12$ .

$- (\frac{1}{12} - \frac{12 \cdot 0}{12 \cdot 1}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.3.2.2

Cancel the common factor.

$- (\frac{1}{12} - \frac{12 \cdot 0}{12 \cdot 1}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.3.2.3

Rewrite the expression.

$- (\frac{1}{12} - \frac{0}{1}) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.3.2.4

Divide $0$ by $1$ .

$- (\frac{1}{12} - 0) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.4

Multiply $- 1$ by $0$ .

$- (\frac{1}{12} + 0) + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.5

Add $\frac{1}{12}$ and $0$ .

$- \frac{1}{12} + 3 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.6

One to any power is one.

$- \frac{1}{12} + 3 (\frac{1}{10} - \frac{0^{10}}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.7

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{12} + 3 (\frac{1}{10} - \frac{0}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.8

Cancel the common factor of $0$ and $10$ .

Tap for more steps...

Step 17.5.8.1

Factor $10$ out of $0$ .

$- \frac{1}{12} + 3 (\frac{1}{10} - \frac{10 (0)}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.8.2

Cancel the common factors.

Tap for more steps...

Step 17.5.8.2.1

Factor $10$ out of $10$ .

$- \frac{1}{12} + 3 (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.8.2.2

Cancel the common factor.

$- \frac{1}{12} + 3 (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.8.2.3

Rewrite the expression.

$- \frac{1}{12} + 3 (\frac{1}{10} - \frac{0}{1}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.8.2.4

Divide $0$ by $1$ .

$- \frac{1}{12} + 3 (\frac{1}{10} - 0) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.9

Multiply $- 1$ by $0$ .

$- \frac{1}{12} + 3 (\frac{1}{10} + 0) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.10

Add $\frac{1}{10}$ and $0$ .

$- \frac{1}{12} + 3 (\frac{1}{10}) - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.11

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

$- \frac{1}{12} + \frac{3}{10} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.12

To write $- \frac{1}{12}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{1}{12} \cdot \frac{5}{5} + \frac{3}{10} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.13

To write $\frac{3}{10}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{1}{12} \cdot \frac{5}{5} + \frac{3}{10} \cdot \frac{6}{6} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.14

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 17.5.14.1

Multiply $\frac{1}{12}$ by $\frac{5}{5}$ .

$- \frac{5}{12 \cdot 5} + \frac{3}{10} \cdot \frac{6}{6} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.14.2

Multiply $12$ by $5$ .

$- \frac{5}{60} + \frac{3}{10} \cdot \frac{6}{6} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.14.3

Multiply $\frac{3}{10}$ by $\frac{6}{6}$ .

$- \frac{5}{60} + \frac{3 \cdot 6}{10 \cdot 6} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.14.4

Multiply $10$ by $6$ .

$- \frac{5}{60} + \frac{3 \cdot 6}{60} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.15

Combine the numerators over the common denominator.

$\frac{- 5 + 3 \cdot 6}{60} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.16

Simplify the numerator.

Tap for more steps...

Step 17.5.16.1

Multiply $3$ by $6$ .

$\frac{- 5 + 18}{60} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.16.2

Add $- 5$ and $18$ .

$\frac{13}{60} - 3 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.17

One to any power is one.

$\frac{13}{60} - 3 (\frac{1}{8} - \frac{0^{8}}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.18

Raising $0$ to any positive power yields $0$ .

$\frac{13}{60} - 3 (\frac{1}{8} - \frac{0}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.19

Cancel the common factor of $0$ and $8$ .

Tap for more steps...

Step 17.5.19.1

Factor $8$ out of $0$ .

$\frac{13}{60} - 3 (\frac{1}{8} - \frac{8 (0)}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.19.2

Cancel the common factors.

Tap for more steps...

Step 17.5.19.2.1

Factor $8$ out of $8$ .

$\frac{13}{60} - 3 (\frac{1}{8} - \frac{8 \cdot 0}{8 \cdot 1}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.19.2.2

Cancel the common factor.

$\frac{13}{60} - 3 (\frac{1}{8} - \frac{8 \cdot 0}{8 \cdot 1}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.19.2.3

Rewrite the expression.

$\frac{13}{60} - 3 (\frac{1}{8} - \frac{0}{1}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.19.2.4

Divide $0$ by $1$ .

$\frac{13}{60} - 3 (\frac{1}{8} - 0) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.20

Multiply $- 1$ by $0$ .

$\frac{13}{60} - 3 (\frac{1}{8} + 0) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.21

Add $\frac{1}{8}$ and $0$ .

$\frac{13}{60} - 3 (\frac{1}{8}) + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.22

Combine $- 3$ and $\frac{1}{8}$ .

$\frac{13}{60} + \frac{- 3}{8} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.23

Move the negative in front of the fraction.

$\frac{13}{60} - \frac{3}{8} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.24

To write $\frac{13}{60}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{13}{60} \cdot \frac{2}{2} - \frac{3}{8} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.25

To write $- \frac{3}{8}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{13}{60} \cdot \frac{2}{2} - \frac{3}{8} \cdot \frac{15}{15} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.26

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

Tap for more steps...

Step 17.5.26.1

Multiply $\frac{13}{60}$ by $\frac{2}{2}$ .

$\frac{13 \cdot 2}{60 \cdot 2} - \frac{3}{8} \cdot \frac{15}{15} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.26.2

Multiply $60$ by $2$ .

$\frac{13 \cdot 2}{120} - \frac{3}{8} \cdot \frac{15}{15} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.26.3

Multiply $\frac{3}{8}$ by $\frac{15}{15}$ .

$\frac{13 \cdot 2}{120} - \frac{3 \cdot 15}{8 \cdot 15} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.26.4

Multiply $8$ by $15$ .

$\frac{13 \cdot 2}{120} - \frac{3 \cdot 15}{120} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.27

Combine the numerators over the common denominator.

$\frac{13 \cdot 2 - 3 \cdot 15}{120} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.28

Simplify the numerator.

Tap for more steps...

Step 17.5.28.1

Multiply $13$ by $2$ .

$\frac{26 - 3 \cdot 15}{120} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.28.2

Multiply $- 3$ by $15$ .

$\frac{26 - 45}{120} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.28.3

Subtract $45$ from $26$ .

$\frac{- 19}{120} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.29

Move the negative in front of the fraction.

$- \frac{19}{120} + \frac{1}{6} \cdot 1^{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.30

One to any power is one.

$- \frac{19}{120} + \frac{1}{6} \cdot 1 - \frac{1}{6} \cdot 0^{6}$

Step 17.5.31

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

$- \frac{19}{120} + \frac{1}{6} - \frac{1}{6} \cdot 0^{6}$

Step 17.5.32

Raising $0$ to any positive power yields $0$ .

$- \frac{19}{120} + \frac{1}{6} - \frac{1}{6} \cdot 0$

Step 17.5.33

Multiply $0$ by $- 1$ .

$- \frac{19}{120} + \frac{1}{6} + 0 (\frac{1}{6})$

Step 17.5.34

Multiply $0$ by $\frac{1}{6}$ .

$- \frac{19}{120} + \frac{1}{6} + 0$

Step 17.5.35

Add $\frac{1}{6}$ and $0$ .

$- \frac{19}{120} + \frac{1}{6}$

Step 17.5.36

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$- \frac{19}{120} + \frac{1}{6} \cdot \frac{20}{20}$

Step 17.5.37

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

Tap for more steps...

Step 17.5.37.1

Multiply $\frac{1}{6}$ by $\frac{20}{20}$ .

$- \frac{19}{120} + \frac{20}{6 \cdot 20}$

Step 17.5.37.2

Multiply $6$ by $20$ .

$- \frac{19}{120} + \frac{20}{120}$

Step 17.5.38

Combine the numerators over the common denominator.

$\frac{- 19 + 20}{120}$

Step 17.5.39

Add $- 19$ and $20$ .

$\frac{1}{120}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{120}$

Decimal Form:

$0.008\overline{3}$