Trigonometry Examples

$\sqrt{\frac{1 - (- \frac{8}{\sqrt{113}})}{2}}$

Step 1

Multiply $\frac{8}{\sqrt{113}}$ by $\frac{\sqrt{113}}{\sqrt{113}}$ .

$\sqrt{\frac{1 - - (\frac{8}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}})}{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{8}{\sqrt{113}}$ by $\frac{\sqrt{113}}{\sqrt{113}}$ .

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{\sqrt{113} \sqrt{113}}}{2}}$

Step 2.2

Raise $\sqrt{113}$ to the power of $1$ .

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{{\sqrt{113}}^{1} \sqrt{113}}}{2}}$

Step 2.3

Raise $\sqrt{113}$ to the power of $1$ .

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{{\sqrt{113}}^{1} {\sqrt{113}}^{1}}}{2}}$

Step 2.4

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

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{{\sqrt{113}}^{1 + 1}}}{2}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{{\sqrt{113}}^{2}}}{2}}$

Step 2.6

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

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{113}$ as $113^{\frac{1}{2}}$ .

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}}{2}}$

Step 2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}}{2}}$

Step 2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{113^{\frac{2}{2}}}}{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{113^{\frac{2}{2}}}}{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{113^{1}}}{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1 - - \frac{8 \sqrt{113}}{113}}{2}}$

Step 3

Multiply $- - \frac{8 \sqrt{113}}{113}$ .

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Step 3.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{1 + 1 \frac{8 \sqrt{113}}{113}}{2}}$

Step 3.2

Multiply $\frac{8 \sqrt{113}}{113}$ by $1$ .

$\sqrt{\frac{1 + \frac{8 \sqrt{113}}{113}}{2}}$

Step 4

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{113}{113} + \frac{8 \sqrt{113}}{113}}{2}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{113 + 8 \sqrt{113}}{113}}{2}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{113 + 8 \sqrt{113}}{113} \cdot \frac{1}{2}}$

Step 7

Multiply $\frac{113 + 8 \sqrt{113}}{113} \cdot \frac{1}{2}$ .

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Step 7.1

Multiply $\frac{113 + 8 \sqrt{113}}{113}$ by $\frac{1}{2}$ .

$\sqrt{\frac{113 + 8 \sqrt{113}}{113 \cdot 2}}$

Step 7.2

Multiply $113$ by $2$ .

$\sqrt{\frac{113 + 8 \sqrt{113}}{226}}$

Step 8

Rewrite $\sqrt{\frac{113 + 8 \sqrt{113}}{226}}$ as $\frac{\sqrt{113 + 8 \sqrt{113}}}{\sqrt{226}}$ .

$\frac{\sqrt{113 + 8 \sqrt{113}}}{\sqrt{226}}$

Step 9

Multiply $\frac{\sqrt{113 + 8 \sqrt{113}}}{\sqrt{226}}$ by $\frac{\sqrt{226}}{\sqrt{226}}$ .

$\frac{\sqrt{113 + 8 \sqrt{113}}}{\sqrt{226}} \cdot \frac{\sqrt{226}}{\sqrt{226}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{\sqrt{113 + 8 \sqrt{113}}}{\sqrt{226}}$ by $\frac{\sqrt{226}}{\sqrt{226}}$ .

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{\sqrt{226} \sqrt{226}}$

Step 10.2

Raise $\sqrt{226}$ to the power of $1$ .

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{{\sqrt{226}}^{1} \sqrt{226}}$

Step 10.3

Raise $\sqrt{226}$ to the power of $1$ .

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{{\sqrt{226}}^{1} {\sqrt{226}}^{1}}$

Step 10.4

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

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{{\sqrt{226}}^{1 + 1}}$

Step 10.5

Add $1$ and $1$ .

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{{\sqrt{226}}^{2}}$

Step 10.6

Rewrite ${\sqrt{226}}^{2}$ as $226$ .

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Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{226}$ as $226^{\frac{1}{2}}$ .

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{{(226^{\frac{1}{2}})}^{2}}$

Step 10.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{226^{\frac{1}{2} \cdot 2}}$

Step 10.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{226^{\frac{2}{2}}}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{226^{\frac{2}{2}}}$

Step 10.6.4.2

Rewrite the expression.

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{226^{1}}$

Step 10.6.5

Evaluate the exponent.

$\frac{\sqrt{113 + 8 \sqrt{113}} \sqrt{226}}{226}$

Step 11

Combine using the product rule for radicals.

$\frac{\sqrt{(113 + 8 \sqrt{113}) \cdot 226}}{226}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(113 + 8 \sqrt{113}) \cdot 226}}{226}$

Decimal Form:

$0.93610274 \dots$