Trigonometry Examples

\sqrt{{(- \frac{1}{6})}^{2} + {(- \frac{1}{8})}^{2}}

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

Step 1

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to

- \frac{1}{6}

$- \frac{1}{6}$ .

\sqrt{{(- 1)}^{2} {(\frac{1}{6})}^{2} + {(- \frac{1}{8})}^{2}}

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

Step 1.2

Apply the product rule to

\frac{1}{6}

$\frac{1}{6}$ .

\sqrt{{(- 1)}^{2} \frac{1^{2}}{6^{2}} + {(- \frac{1}{8})}^{2}}

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

\sqrt{{(- 1)}^{2} \frac{1^{2}}{6^{2}} + {(- \frac{1}{8})}^{2}}

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

Step 2

Raise

- 1

$- 1$ to the power of

2

$2$ .

\sqrt{1 \frac{1^{2}}{6^{2}} + {(- \frac{1}{8})}^{2}}

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

Step 3

Multiply

\frac{1^{2}}{6^{2}}

$\frac{1^{2}}{6^{2}}$ by

1

$1$ .

\sqrt{\frac{1^{2}}{6^{2}} + {(- \frac{1}{8})}^{2}}

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

Step 4

One to any power is one.

\sqrt{\frac{1}{6^{2}} + {(- \frac{1}{8})}^{2}}

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

Step 5

Raise

6

$6$ to the power of

2

$2$ .

\sqrt{\frac{1}{36} + {(- \frac{1}{8})}^{2}}

$\sqrt{\frac{1}{36} + {(- \frac{1}{8})}^{2}}$

Step 6

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.1

Apply the product rule to

- \frac{1}{8}

$- \frac{1}{8}$ .

\sqrt{\frac{1}{36} + {(- 1)}^{2} {(\frac{1}{8})}^{2}}

$\sqrt{\frac{1}{36} + {(- 1)}^{2} {(\frac{1}{8})}^{2}}$

Step 6.2

Apply the product rule to

\frac{1}{8}

$\frac{1}{8}$ .

\sqrt{\frac{1}{36} + {(- 1)}^{2} \frac{1^{2}}{8^{2}}}

$\sqrt{\frac{1}{36} + {(- 1)}^{2} \frac{1^{2}}{8^{2}}}$

\sqrt{\frac{1}{36} + {(- 1)}^{2} \frac{1^{2}}{8^{2}}}

$\sqrt{\frac{1}{36} + {(- 1)}^{2} \frac{1^{2}}{8^{2}}}$

Step 7

Raise

- 1

$- 1$ to the power of

2

$2$ .

\sqrt{\frac{1}{36} + 1 \frac{1^{2}}{8^{2}}}

$\sqrt{\frac{1}{36} + 1 \frac{1^{2}}{8^{2}}}$

Step 8

Multiply

\frac{1^{2}}{8^{2}}

$\frac{1^{2}}{8^{2}}$ by

1

$1$ .

\sqrt{\frac{1}{36} + \frac{1^{2}}{8^{2}}}

$\sqrt{\frac{1}{36} + \frac{1^{2}}{8^{2}}}$

Step 9

One to any power is one.

\sqrt{\frac{1}{36} + \frac{1}{8^{2}}}

$\sqrt{\frac{1}{36} + \frac{1}{8^{2}}}$

Step 10

Raise

8

$8$ to the power of

2

$2$ .

\sqrt{\frac{1}{36} + \frac{1}{64}}

$\sqrt{\frac{1}{36} + \frac{1}{64}}$

Step 11

To write

\frac{1}{36}

$\frac{1}{36}$ as a fraction with a common denominator, multiply by

\frac{16}{16}

$\frac{16}{16}$ .

\sqrt{\frac{1}{36} \cdot \frac{16}{16} + \frac{1}{64}}

$\sqrt{\frac{1}{36} \cdot \frac{16}{16} + \frac{1}{64}}$

Step 12

To write

\frac{1}{64}

$\frac{1}{64}$ as a fraction with a common denominator, multiply by

\frac{9}{9}

$\frac{9}{9}$ .

\sqrt{\frac{1}{36} \cdot \frac{16}{16} + \frac{1}{64} \cdot \frac{9}{9}}

$\sqrt{\frac{1}{36} \cdot \frac{16}{16} + \frac{1}{64} \cdot \frac{9}{9}}$

Step 13

Write each expression with a common denominator of

576

$576$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 13.1

Multiply

\frac{1}{36}

$\frac{1}{36}$ by

\frac{16}{16}

$\frac{16}{16}$ .

\sqrt{\frac{16}{36 \cdot 16} + \frac{1}{64} \cdot \frac{9}{9}}

$\sqrt{\frac{16}{36 \cdot 16} + \frac{1}{64} \cdot \frac{9}{9}}$

Step 13.2

Multiply

36

$36$ by

16

$16$ .

\sqrt{\frac{16}{576} + \frac{1}{64} \cdot \frac{9}{9}}

$\sqrt{\frac{16}{576} + \frac{1}{64} \cdot \frac{9}{9}}$

Step 13.3

Multiply

\frac{1}{64}

$\frac{1}{64}$ by

\frac{9}{9}

$\frac{9}{9}$ .

\sqrt{\frac{16}{576} + \frac{9}{64 \cdot 9}}

$\sqrt{\frac{16}{576} + \frac{9}{64 \cdot 9}}$

Step 13.4

Multiply

64

$64$ by

9

$9$ .

\sqrt{\frac{16}{576} + \frac{9}{576}}

$\sqrt{\frac{16}{576} + \frac{9}{576}}$

\sqrt{\frac{16}{576} + \frac{9}{576}}

$\sqrt{\frac{16}{576} + \frac{9}{576}}$

Step 14

Combine the numerators over the common denominator.

\sqrt{\frac{16 + 9}{576}}

$\sqrt{\frac{16 + 9}{576}}$

Step 15

Add

16

$16$ and

9

$9$ .

\sqrt{\frac{25}{576}}

$\sqrt{\frac{25}{576}}$

Step 16

Rewrite

\sqrt{\frac{25}{576}}

$\sqrt{\frac{25}{576}}$ as

\frac{\sqrt{25}}{\sqrt{576}}

$\frac{\sqrt{25}}{\sqrt{576}}$ .

\frac{\sqrt{25}}{\sqrt{576}}

$\frac{\sqrt{25}}{\sqrt{576}}$

Step 17

Simplify the numerator.

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Step 17.1

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

\frac{\sqrt{5^{2}}}{\sqrt{576}}

$\frac{\sqrt{5^{2}}}{\sqrt{576}}$

Step 17.2

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

\frac{5}{\sqrt{576}}

$\frac{5}{\sqrt{576}}$

\frac{5}{\sqrt{576}}

$\frac{5}{\sqrt{576}}$

Step 18

Simplify the denominator.

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Step 18.1

Rewrite

576

$576$ as

24^{2}

$24^{2}$ .

\frac{5}{\sqrt{24^{2}}}

$\frac{5}{\sqrt{24^{2}}}$

Step 18.2

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

\frac{5}{24}

$\frac{5}{24}$

\frac{5}{24}

$\frac{5}{24}$

Step 19

The result can be shown in multiple forms.

Exact Form:

\frac{5}{24}

$\frac{5}{24}$

Decimal Form:

0.208 ¯ 3

$0.208\overline{3}$