Trigonometry Examples

\sqrt{\frac{1 - (\frac{- 6}{5})}{1 + \frac{- 5}{6}}}

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

Step 1

Move the negative in front of the fraction.

\sqrt{\frac{1 - - \frac{6}{5}}{1 + \frac{- 5}{6}}}

$\sqrt{\frac{1 - - \frac{6}{5}}{1 + \frac{- 5}{6}}}$

Step 2

Multiply

- - \frac{6}{5}

$- - \frac{6}{5}$ .

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Step 2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\sqrt{\frac{1 + 1 (\frac{6}{5})}{1 + \frac{- 5}{6}}}

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

Step 2.2

Multiply

\frac{6}{5}

$\frac{6}{5}$ by

1

$1$ .

\sqrt{\frac{1 + \frac{6}{5}}{1 + \frac{- 5}{6}}}

$\sqrt{\frac{1 + \frac{6}{5}}{1 + \frac{- 5}{6}}}$

\sqrt{\frac{1 + \frac{6}{5}}{1 + \frac{- 5}{6}}}

$\sqrt{\frac{1 + \frac{6}{5}}{1 + \frac{- 5}{6}}}$

Step 3

Write

1

$1$ as a fraction with a common denominator.

\sqrt{\frac{\frac{5}{5} + \frac{6}{5}}{1 + \frac{- 5}{6}}}

$\sqrt{\frac{\frac{5}{5} + \frac{6}{5}}{1 + \frac{- 5}{6}}}$

Step 4

Combine the numerators over the common denominator.

\sqrt{\frac{\frac{5 + 6}{5}}{1 + \frac{- 5}{6}}}

$\sqrt{\frac{\frac{5 + 6}{5}}{1 + \frac{- 5}{6}}}$

Step 5

Add

5

$5$ and

6

$6$ .

\sqrt{\frac{\frac{11}{5}}{1 + \frac{- 5}{6}}}

$\sqrt{\frac{\frac{11}{5}}{1 + \frac{- 5}{6}}}$

Step 6

Write

1

$1$ as a fraction with a common denominator.

\sqrt{\frac{\frac{11}{5}}{\frac{6}{6} + \frac{- 5}{6}}}

$\sqrt{\frac{\frac{11}{5}}{\frac{6}{6} + \frac{- 5}{6}}}$

Step 7

Combine the numerators over the common denominator.

\sqrt{\frac{\frac{11}{5}}{\frac{6 - 5}{6}}}

$\sqrt{\frac{\frac{11}{5}}{\frac{6 - 5}{6}}}$

Step 8

Subtract

5

$5$ from

6

$6$ .

\sqrt{\frac{\frac{11}{5}}{\frac{1}{6}}}

$\sqrt{\frac{\frac{11}{5}}{\frac{1}{6}}}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

\sqrt{\frac{11}{5} \cdot 6}

$\sqrt{\frac{11}{5} \cdot 6}$

Step 10

Multiply

\frac{11}{5} \cdot 6

$\frac{11}{5} \cdot 6$ .

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Step 10.1

Combine

\frac{11}{5}

$\frac{11}{5}$ and

6

$6$ .

\sqrt{\frac{11 \cdot 6}{5}}

$\sqrt{\frac{11 \cdot 6}{5}}$

Step 10.2

Multiply

11

$11$ by

6

$6$ .

\sqrt{\frac{66}{5}}

$\sqrt{\frac{66}{5}}$

\sqrt{\frac{66}{5}}

$\sqrt{\frac{66}{5}}$

Step 11

Rewrite

\sqrt{\frac{66}{5}}

$\sqrt{\frac{66}{5}}$ as

\frac{\sqrt{66}}{\sqrt{5}}

$\frac{\sqrt{66}}{\sqrt{5}}$ .

\frac{\sqrt{66}}{\sqrt{5}}

$\frac{\sqrt{66}}{\sqrt{5}}$

Step 12

Multiply

\frac{\sqrt{66}}{\sqrt{5}}

$\frac{\sqrt{66}}{\sqrt{5}}$ by

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

$\frac{\sqrt{5}}{\sqrt{5}}$ .

\frac{\sqrt{66}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}

$\frac{\sqrt{66}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply

\frac{\sqrt{66}}{\sqrt{5}}

$\frac{\sqrt{66}}{\sqrt{5}}$ by

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

$\frac{\sqrt{5}}{\sqrt{5}}$ .

\frac{\sqrt{66} \sqrt{5}}{\sqrt{5} \sqrt{5}}

$\frac{\sqrt{66} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 13.2

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}

$\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 13.3

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}

$\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 13.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}

$\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 13.5

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{2}}

$\frac{\sqrt{66} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 13.6

Rewrite

{\sqrt{5}}^{2}

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

5

$5$ .

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Step 13.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{5}

$\sqrt{5}$ as

5^{\frac{1}{2}}

$5^{\frac{1}{2}}$ .

\frac{\sqrt{66} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}

$\frac{\sqrt{66} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 13.6.2

Apply the power rule and multiply exponents,

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

${(a^{m})}^{n} = a^{m n}$ .

\frac{\sqrt{66} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}

$\frac{\sqrt{66} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 13.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{66} \sqrt{5}}{5^{\frac{2}{2}}}

$\frac{\sqrt{66} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 13.6.4

Cancel the common factor of

2

$2$ .

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Step 13.6.4.1

Cancel the common factor.

$\frac{\sqrt{66} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 13.6.4.2

Rewrite the expression.

$\frac{\sqrt{66} \sqrt{5}}{5^{1}}$

Step 13.6.5

Evaluate the exponent.

$\frac{\sqrt{66} \sqrt{5}}{5}$

Step 14

Simplify the numerator.

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Step 14.1

Combine using the product rule for radicals.

$\frac{\sqrt{66 \cdot 5}}{5}$

Step 14.2

Multiply

$66$ by

$5$ .

$\frac{\sqrt{330}}{5}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{330}}{5}$

Decimal Form:

$3.63318042 \dots$