Trigonometry Examples

\sqrt{\frac{\sqrt{113} - 7}{\frac{\sqrt{113}}{2}}}

$\sqrt{\frac{\sqrt{113} - 7}{\frac{\sqrt{113}}{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

\sqrt{(\sqrt{113} - 7) \frac{2}{\sqrt{113}}}

$\sqrt{(\sqrt{113} - 7) \frac{2}{\sqrt{113}}}$

Step 2

Multiply

\frac{2}{\sqrt{113}}

$\frac{2}{\sqrt{113}}$ by

\frac{\sqrt{113}}{\sqrt{113}}

$\frac{\sqrt{113}}{\sqrt{113}}$ .

\sqrt{(\sqrt{113} - 7) (\frac{2}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}})}

$\sqrt{(\sqrt{113} - 7) (\frac{2}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}})}$

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

Multiply

\frac{2}{\sqrt{113}}

$\frac{2}{\sqrt{113}}$ by

\frac{\sqrt{113}}{\sqrt{113}}

$\frac{\sqrt{113}}{\sqrt{113}}$ .

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{\sqrt{113} \sqrt{113}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{\sqrt{113} \sqrt{113}}}$

Step 3.2

Raise

\sqrt{113}

$\sqrt{113}$ to the power of

1

$1$ .

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{1} \sqrt{113}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{1} \sqrt{113}}}$

Step 3.3

Raise

\sqrt{113}

$\sqrt{113}$ to the power of

1

$1$ .

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{1} {\sqrt{113}}^{1}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{1} {\sqrt{113}}^{1}}}$

Step 3.4

Use the power rule

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

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

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{1 + 1}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{1 + 1}}}$

Step 3.5

Add

1

$1$ and

1

$1$ .

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{2}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{\sqrt{113}}^{2}}}$

Step 3.6

Rewrite

{\sqrt{113}}^{2}

${\sqrt{113}}^{2}$ as

113

$113$ .

Tap for more steps...

Step 3.6.1

Use

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

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

\sqrt{113}

$\sqrt{113}$ as

113^{\frac{1}{2}}

$113^{\frac{1}{2}}$ .

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}}$

Step 3.6.2

Apply the power rule and multiply exponents,

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

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

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}}$

Step 3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{\frac{2}{2}}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{\frac{2}{2}}}}$

Step 3.6.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 3.6.4.1

Cancel the common factor.

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{\frac{2}{2}}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{\frac{2}{2}}}}$

Step 3.6.4.2

Rewrite the expression.

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{1}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{1}}}$

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{1}}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113^{1}}}$

Step 3.6.5

Evaluate the exponent.

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113}}$

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113}}$

\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113}}

$\sqrt{(\sqrt{113} - 7) \frac{2 \sqrt{113}}{113}}$

Step 4

Apply the distributive property.

\sqrt{\sqrt{113} \frac{2 \sqrt{113}}{113} - 7 \frac{2 \sqrt{113}}{113}}

$\sqrt{\sqrt{113} \frac{2 \sqrt{113}}{113} - 7 \frac{2 \sqrt{113}}{113}}$

Step 5

Multiply

\sqrt{113} \frac{2 \sqrt{113}}{113}

$\sqrt{113} \frac{2 \sqrt{113}}{113}$ .

Tap for more steps...

Step 5.1

Combine

\sqrt{113}

$\sqrt{113}$ and

\frac{2 \sqrt{113}}{113}

$\frac{2 \sqrt{113}}{113}$ .

\sqrt{\frac{\sqrt{113} (2 \sqrt{113})}{113} - 7 \frac{2 \sqrt{113}}{113}}

$\sqrt{\frac{\sqrt{113} (2 \sqrt{113})}{113} - 7 \frac{2 \sqrt{113}}{113}}$

Step 5.2

Raise

\sqrt{113}

$\sqrt{113}$ to the power of

1

$1$ .

\sqrt{\frac{2 ({\sqrt{113}}^{1} \sqrt{113})}{113} - 7 \frac{2 \sqrt{113}}{113}}

$\sqrt{\frac{2 ({\sqrt{113}}^{1} \sqrt{113})}{113} - 7 \frac{2 \sqrt{113}}{113}}$

Step 5.3

Raise

\sqrt{113}

$\sqrt{113}$ to the power of

1

$1$ .

\sqrt{\frac{2 ({\sqrt{113}}^{1} {\sqrt{113}}^{1})}{113} - 7 \frac{2 \sqrt{113}}{113}}

$\sqrt{\frac{2 ({\sqrt{113}}^{1} {\sqrt{113}}^{1})}{113} - 7 \frac{2 \sqrt{113}}{113}}$

Step 5.4

Use the power rule

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

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

\sqrt{\frac{2 {\sqrt{113}}^{1 + 1}}{113} - 7 \frac{2 \sqrt{113}}{113}}

$\sqrt{\frac{2 {\sqrt{113}}^{1 + 1}}{113} - 7 \frac{2 \sqrt{113}}{113}}$

Step 5.5

Add

1

$1$ and

1

$1$ .

\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} - 7 \frac{2 \sqrt{113}}{113}}

$\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} - 7 \frac{2 \sqrt{113}}{113}}$

\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} - 7 \frac{2 \sqrt{113}}{113}}

$\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} - 7 \frac{2 \sqrt{113}}{113}}$

Step 6

Multiply

- 7 \frac{2 \sqrt{113}}{113}

$- 7 \frac{2 \sqrt{113}}{113}$ .

Tap for more steps...

Step 6.1

Combine

- 7

$- 7$ and

\frac{2 \sqrt{113}}{113}

$\frac{2 \sqrt{113}}{113}$ .

\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} + \frac{- 7 (2 \sqrt{113})}{113}}

$\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} + \frac{- 7 (2 \sqrt{113})}{113}}$

Step 6.2

Multiply

2

$2$ by

- 7

$- 7$ .

\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} + \frac{- 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} + \frac{- 14 \sqrt{113}}{113}}$

\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} + \frac{- 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 {\sqrt{113}}^{2}}{113} + \frac{- 14 \sqrt{113}}{113}}$

Step 7

Combine the numerators over the common denominator.

\sqrt{\frac{2 {\sqrt{113}}^{2} - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 {\sqrt{113}}^{2} - 14 \sqrt{113}}{113}}$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Rewrite

{\sqrt{113}}^{2}

${\sqrt{113}}^{2}$ as

113

$113$ .

Tap for more steps...

Step 8.1.1

Use

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

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

\sqrt{113}

$\sqrt{113}$ as

113^{\frac{1}{2}}

$113^{\frac{1}{2}}$ .

\sqrt{\frac{2 {(113^{\frac{1}{2}})}^{2} - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 {(113^{\frac{1}{2}})}^{2} - 14 \sqrt{113}}{113}}$

Step 8.1.2

Apply the power rule and multiply exponents,

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

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

\sqrt{\frac{2 \cdot 113^{\frac{1}{2} \cdot 2} - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 \cdot 113^{\frac{1}{2} \cdot 2} - 14 \sqrt{113}}{113}}$

Step 8.1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{2 \cdot 113^{\frac{2}{2}} - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 \cdot 113^{\frac{2}{2}} - 14 \sqrt{113}}{113}}$

Step 8.1.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 8.1.4.1

Cancel the common factor.

\sqrt{\frac{2 \cdot 113^{\frac{2}{2}} - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 \cdot 113^{\frac{2}{2}} - 14 \sqrt{113}}{113}}$

Step 8.1.4.2

Rewrite the expression.

\sqrt{\frac{2 \cdot 113^{1} - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 \cdot 113^{1} - 14 \sqrt{113}}{113}}$

\sqrt{\frac{2 \cdot 113^{1} - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 \cdot 113^{1} - 14 \sqrt{113}}{113}}$

Step 8.1.5

Evaluate the exponent.

\sqrt{\frac{2 \cdot 113 - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 \cdot 113 - 14 \sqrt{113}}{113}}$

\sqrt{\frac{2 \cdot 113 - 14 \sqrt{113}}{113}}

$\sqrt{\frac{2 \cdot 113 - 14 \sqrt{113}}{113}}$

Step 8.2

Multiply

2

$2$ by

113

$113$ .

\sqrt{\frac{226 - 14 \sqrt{113}}{113}}

$\sqrt{\frac{226 - 14 \sqrt{113}}{113}}$

\sqrt{\frac{226 - 14 \sqrt{113}}{113}}

$\sqrt{\frac{226 - 14 \sqrt{113}}{113}}$

Step 9

Rewrite

\sqrt{\frac{226 - 14 \sqrt{113}}{113}}

$\sqrt{\frac{226 - 14 \sqrt{113}}{113}}$ as

\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}

$\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}$ .

\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}

$\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}$

Step 10

Multiply

\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}

$\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}$ by

\frac{\sqrt{113}}{\sqrt{113}}

$\frac{\sqrt{113}}{\sqrt{113}}$ .

\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}}

$\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}}$

Step 11

Combine and simplify the denominator.

Tap for more steps...

Step 11.1

Multiply

\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}

$\frac{\sqrt{226 - 14 \sqrt{113}}}{\sqrt{113}}$ by

\frac{\sqrt{113}}{\sqrt{113}}

$\frac{\sqrt{113}}{\sqrt{113}}$ .

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{\sqrt{113} \sqrt{113}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{\sqrt{113} \sqrt{113}}$

Step 11.2

Raise

\sqrt{113}

$\sqrt{113}$ to the power of

1

$1$ .

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1} \sqrt{113}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1} \sqrt{113}}$

Step 11.3

Raise

\sqrt{113}

$\sqrt{113}$ to the power of

1

$1$ .

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1} {\sqrt{113}}^{1}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1} {\sqrt{113}}^{1}}$

Step 11.4

Use the power rule

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

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

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1 + 1}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1 + 1}}$

Step 11.5

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{2}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{2}}$

Step 11.6

Rewrite

{\sqrt{113}}^{2}

${\sqrt{113}}^{2}$ as

113

$113$ .

Tap for more steps...

Step 11.6.1

Use

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

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

\sqrt{113}

$\sqrt{113}$ as

113^{\frac{1}{2}}

$113^{\frac{1}{2}}$ .

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}$

Step 11.6.2

Apply the power rule and multiply exponents,

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

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

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}$

Step 11.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{\frac{2}{2}}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{\frac{2}{2}}}$

Step 11.6.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 11.6.4.1

Cancel the common factor.

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{\frac{2}{2}}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{\frac{2}{2}}}$

Step 11.6.4.2

Rewrite the expression.

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{1}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{1}}$

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{1}}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113^{1}}$

Step 11.6.5

Evaluate the exponent.

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113}$

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113}$

\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113}

$\frac{\sqrt{226 - 14 \sqrt{113}} \sqrt{113}}{113}$

Step 12

Combine using the product rule for radicals.

\frac{\sqrt{(226 - 14 \sqrt{113}) \cdot 113}}{113}

$\frac{\sqrt{(226 - 14 \sqrt{113}) \cdot 113}}{113}$

Step 13

The result can be shown in multiple forms.

Exact Form:

\frac{\sqrt{(226 - 14 \sqrt{113}) \cdot 113}}{113}

$\frac{\sqrt{(226 - 14 \sqrt{113}) \cdot 113}}{113}$

Decimal Form:

0.82643256 \dots

$0.82643256 \dots$