Trigonometry Examples

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

Step 1

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

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Step 1.1

Apply the product rule to $- \frac{1}{6}$ .

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

Step 1.2

Apply the product rule to $\frac{1}{6}$ .

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

Step 2

Raise $- 1$ to the power of $2$ .

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

Step 3

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

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

Step 4

One to any power is one.

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

Step 5

Raise $6$ to the power of $2$ .

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

Step 6

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

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Step 6.1

Apply the product rule to $- \frac{1}{8}$ .

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

Step 6.2

Apply the product rule to $\frac{1}{8}$ .

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

Step 7

Raise $- 1$ to the power of $2$ .

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

Step 8

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

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

Step 9

One to any power is one.

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

Step 10

Raise $8$ to the power of $2$ .

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

Step 11

To write $\frac{1}{36}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

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

Step 12

To write $\frac{1}{64}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 13

Write each expression with a common denominator of $576$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.1

Multiply $\frac{1}{36}$ by $\frac{16}{16}$ .

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

Step 13.2

Multiply $36$ by $16$ .

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

Step 13.3

Multiply $\frac{1}{64}$ by $\frac{9}{9}$ .

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

Step 13.4

Multiply $64$ by $9$ .

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

Step 14

Combine the numerators over the common denominator.

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

Step 15

Add $16$ and $9$ .

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

Step 16

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

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

Step 17

Simplify the numerator.

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Step 17.1

Rewrite $25$ as $5^{2}$ .

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

Step 17.2

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

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

Step 18

Simplify the denominator.

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Step 18.1

Rewrite $576$ as $24^{2}$ .

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

Step 18.2

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

$\frac{5}{24}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{24}$

Decimal Form:

$0.208\overline{3}$