Trigonometry Examples

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

Step 1

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $113$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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

Step 8

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

$\sqrt{(113 - 8 \sqrt{113}) \frac{113 - 8 \sqrt{113}}{(113 + 8 \sqrt{113}) (113 - 8 \sqrt{113})}}$

Step 9

Expand the denominator using the FOIL method.

$\sqrt{(113 - 8 \sqrt{113}) \frac{113 - 8 \sqrt{113}}{12769 - 904 \sqrt{113} + 904 \sqrt{113} - 64 {\sqrt{113}}^{2}}}$

Step 10

Simplify.

$\sqrt{(113 - 8 \sqrt{113}) \frac{113 - 8 \sqrt{113}}{5537}}$

Step 11

Apply the distributive property.

$\sqrt{113 \frac{113 - 8 \sqrt{113}}{5537} - 8 \sqrt{113} \frac{113 - 8 \sqrt{113}}{5537}}$

Step 12

Cancel the common factor of $113$ .

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Step 12.1

Factor $113$ out of $5537$ .

$\sqrt{113 \frac{113 - 8 \sqrt{113}}{113 (49)} - 8 \sqrt{113} \frac{113 - 8 \sqrt{113}}{5537}}$

Step 12.2

Cancel the common factor.

$\sqrt{113 \frac{113 - 8 \sqrt{113}}{113 \cdot 49} - 8 \sqrt{113} \frac{113 - 8 \sqrt{113}}{5537}}$

Step 12.3

Rewrite the expression.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} - 8 \sqrt{113} \frac{113 - 8 \sqrt{113}}{5537}}$

Step 13

Multiply $- 8 \sqrt{113} \frac{113 - 8 \sqrt{113}}{5537}$ .

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Step 13.1

Combine $\frac{113 - 8 \sqrt{113}}{5537}$ and $- 8$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 - 8 \sqrt{113}) \cdot - 8}{5537} \sqrt{113}}$

Step 13.2

Combine $\frac{(113 - 8 \sqrt{113}) \cdot - 8}{5537}$ and $\sqrt{113}$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 - 8 \sqrt{113}) \cdot - 8 \sqrt{113}}{5537}}$

Step 14

Simplify each term.

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Step 14.1

Group $\sqrt{113}$ and $113 - 8 \sqrt{113}$ together.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{\sqrt{113} (113 - 8 \sqrt{113}) \cdot - 8}{5537}}$

Step 14.2

Apply the distributive property.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(\sqrt{113} \cdot 113 + \sqrt{113} (- 8 \sqrt{113})) \cdot - 8}{5537}}$

Step 14.3

Move $113$ to the left of $\sqrt{113}$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \cdot \sqrt{113} + \sqrt{113} (- 8 \sqrt{113})) \cdot - 8}{5537}}$

Step 14.4

Multiply $\sqrt{113} (- 8 \sqrt{113})$ .

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Step 14.4.1

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

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \cdot \sqrt{113} - 8 ({\sqrt{113}}^{1} \sqrt{113})) \cdot - 8}{5537}}$

Step 14.4.2

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

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \cdot \sqrt{113} - 8 ({\sqrt{113}}^{1} {\sqrt{113}}^{1})) \cdot - 8}{5537}}$

Step 14.4.3

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

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \cdot \sqrt{113} - 8 {\sqrt{113}}^{1 + 1}) \cdot - 8}{5537}}$

Step 14.4.4

Add $1$ and $1$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \cdot \sqrt{113} - 8 {\sqrt{113}}^{2}) \cdot - 8}{5537}}$

Step 14.5

Simplify each term.

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Step 14.5.1

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

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Step 14.5.1.1

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

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \sqrt{113} - 8 {(113^{\frac{1}{2}})}^{2}) \cdot - 8}{5537}}$

Step 14.5.1.2

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

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \sqrt{113} - 8 \cdot 113^{\frac{1}{2} \cdot 2}) \cdot - 8}{5537}}$

Step 14.5.1.3

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

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \sqrt{113} - 8 \cdot 113^{\frac{2}{2}}) \cdot - 8}{5537}}$

Step 14.5.1.4

Cancel the common factor of $2$ .

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Step 14.5.1.4.1

Cancel the common factor.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \sqrt{113} - 8 \cdot 113^{\frac{2}{2}}) \cdot - 8}{5537}}$

Step 14.5.1.4.2

Rewrite the expression.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \sqrt{113} - 8 \cdot 113^{1}) \cdot - 8}{5537}}$

Step 14.5.1.5

Evaluate the exponent.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \sqrt{113} - 8 \cdot 113) \cdot - 8}{5537}}$

Step 14.5.2

Multiply $- 8$ by $113$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(113 \sqrt{113} - 904) \cdot - 8}{5537}}$

Step 14.6

Cancel the common factor of $113 \sqrt{113} - 904$ and $5537$ .

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Step 14.6.1

Factor $113$ out of $(113 \sqrt{113} - 904) \cdot - 8$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{113 ((\sqrt{113} - 8) \cdot - 8)}{5537}}$

Step 14.6.2

Cancel the common factors.

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Step 14.6.2.1

Factor $113$ out of $5537$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{113 ((\sqrt{113} - 8) \cdot - 8)}{113 (49)}}$

Step 14.6.2.2

Cancel the common factor.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{113 ((\sqrt{113} - 8) \cdot - 8)}{113 \cdot 49}}$

Step 14.6.2.3

Rewrite the expression.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{(\sqrt{113} - 8) \cdot - 8}{49}}$

Step 14.7

Move $- 8$ to the left of $\sqrt{113} - 8$ .

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} + \frac{- 8 \cdot (\sqrt{113} - 8)}{49}}$

Step 14.8

Move the negative in front of the fraction.

$\sqrt{\frac{113 - 8 \sqrt{113}}{49} - \frac{8 (\sqrt{113} - 8)}{49}}$

Step 15

Combine the numerators over the common denominator.

$\sqrt{\frac{113 - 8 \sqrt{113} - 8 (\sqrt{113} - 8)}{49}}$

Step 16

Simplify each term.

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Step 16.1

Apply the distributive property.

$\sqrt{\frac{113 - 8 \sqrt{113} - 8 \sqrt{113} - 8 \cdot - 8}{49}}$

Step 16.2

Multiply $- 8$ by $- 8$ .

$\sqrt{\frac{113 - 8 \sqrt{113} - 8 \sqrt{113} + 64}{49}}$

Step 17

Simplify by adding terms.

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Step 17.1

Add $113$ and $64$ .

$\sqrt{\frac{177 - 8 \sqrt{113} - 8 \sqrt{113}}{49}}$

Step 17.2

Subtract $8 \sqrt{113}$ from $- 8 \sqrt{113}$ .

$\sqrt{\frac{177 - 16 \sqrt{113}}{49}}$

Step 18

Rewrite $\sqrt{\frac{177 - 16 \sqrt{113}}{49}}$ as $\frac{\sqrt{177 - 16 \sqrt{113}}}{\sqrt{49}}$ .

$\frac{\sqrt{177 - 16 \sqrt{113}}}{\sqrt{49}}$

Step 19

Simplify the denominator.

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Step 19.1

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{177 - 16 \sqrt{113}}}{\sqrt{7^{2}}}$

Step 19.2

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

$\frac{\sqrt{177 - 16 \sqrt{113}}}{7}$

Step 20

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{177 - 16 \sqrt{113}}}{7}$

Decimal Form:

$0.37573511 \dots$