Trigonometry Examples

$\sqrt{\frac{1 - \frac{2}{\sqrt{13}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 1

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sqrt{\frac{1 - (\frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.2

Raise $\sqrt{13}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.3

Raise $\sqrt{13}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.4

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

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{{\sqrt{13}}^{2}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.6.2

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

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.6.3

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

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{13^{1}}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1 - \frac{2 \sqrt{13}}{13}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 3

Simplify the expression.

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Step 3.1

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

$\sqrt{\frac{\frac{13}{13} - \frac{2 \sqrt{13}}{13}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{13 - 2 \sqrt{13}}{13}}{1 + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 3.3

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

$\sqrt{\frac{\frac{13 - 2 \sqrt{13}}{13}}{\frac{\sqrt{13}}{\sqrt{13}} + \frac{\sqrt{2}}{\sqrt{13}}}}$

Step 3.4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{13 - 2 \sqrt{13}}{13}}{\frac{\sqrt{13} + \sqrt{2}}{\sqrt{13}}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13}}{\sqrt{13} + \sqrt{2}}}$

Step 5

Multiply $\frac{\sqrt{13}}{\sqrt{13} + \sqrt{2}}$ by $\frac{\sqrt{13} - \sqrt{2}}{\sqrt{13} - \sqrt{2}}$ .

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} (\frac{\sqrt{13}}{\sqrt{13} + \sqrt{2}} \cdot \frac{\sqrt{13} - \sqrt{2}}{\sqrt{13} - \sqrt{2}})}$

Step 6

Multiply $\frac{\sqrt{13}}{\sqrt{13} + \sqrt{2}}$ by $\frac{\sqrt{13} - \sqrt{2}}{\sqrt{13} - \sqrt{2}}$ .

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13} (\sqrt{13} - \sqrt{2})}{(\sqrt{13} + \sqrt{2}) (\sqrt{13} - \sqrt{2})}}$

Step 7

Expand the denominator using the FOIL method.

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13} (\sqrt{13} - \sqrt{2})}{{\sqrt{13}}^{2} - \sqrt{26} + \sqrt{26} - {\sqrt{2}}^{2}}}$

Step 8

Simplify.

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13} (\sqrt{13} - \sqrt{2})}{11}}$

Step 9

Apply the distributive property.

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13} \sqrt{13} + \sqrt{13} (- \sqrt{2})}{11}}$

Step 10

Combine using the product rule for radicals.

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13 \cdot 13} + \sqrt{13} (- \sqrt{2})}{11}}$

Step 11

Multiply $\sqrt{13} (- \sqrt{2})$ .

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Step 11.1

Combine using the product rule for radicals.

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13 \cdot 13} - \sqrt{13 \cdot 2}}{11}}$

Step 11.2

Multiply $13$ by $2$ .

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13 \cdot 13} - \sqrt{26}}{11}}$

Step 12

Simplify each term.

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Step 12.1

Multiply $13$ by $13$ .

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{169} - \sqrt{26}}{11}}$

Step 12.2

Rewrite $169$ as $13^{2}$ .

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{\sqrt{13^{2}} - \sqrt{26}}{11}}$

Step 12.3

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

$\sqrt{\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{13 - \sqrt{26}}{11}}$

Step 13

Multiply $\frac{13 - 2 \sqrt{13}}{13} \cdot \frac{13 - \sqrt{26}}{11}$ .

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Step 13.1

Multiply $\frac{13 - 2 \sqrt{13}}{13}$ by $\frac{13 - \sqrt{26}}{11}$ .

$\sqrt{\frac{(13 - 2 \sqrt{13}) (13 - \sqrt{26})}{13 \cdot 11}}$

Step 13.2

Multiply $13$ by $11$ .

$\sqrt{\frac{(13 - 2 \sqrt{13}) (13 - \sqrt{26})}{143}}$

Step 14

Expand $(13 - 2 \sqrt{13}) (13 - \sqrt{26})$ using the FOIL Method.

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Step 14.1

Apply the distributive property.

$\sqrt{\frac{13 (13 - \sqrt{26}) - 2 \sqrt{13} (13 - \sqrt{26})}{143}}$

Step 14.2

Apply the distributive property.

$\sqrt{\frac{13 \cdot 13 + 13 (- \sqrt{26}) - 2 \sqrt{13} (13 - \sqrt{26})}{143}}$

Step 14.3

Apply the distributive property.

$\sqrt{\frac{13 \cdot 13 + 13 (- \sqrt{26}) - 2 \sqrt{13} \cdot 13 - 2 \sqrt{13} (- \sqrt{26})}{143}}$

Step 15

Simplify terms.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $13$ by $13$ .

$\sqrt{\frac{169 + 13 (- \sqrt{26}) - 2 \sqrt{13} \cdot 13 - 2 \sqrt{13} (- \sqrt{26})}{143}}$

Step 15.1.2

Multiply $- 1$ by $13$ .

$\sqrt{\frac{169 - 13 \sqrt{26} - 2 \sqrt{13} \cdot 13 - 2 \sqrt{13} (- \sqrt{26})}{143}}$

Step 15.1.3

Multiply $13$ by $- 2$ .

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} - 2 \sqrt{13} (- \sqrt{26})}{143}}$

Step 15.1.4

Multiply $- 2 \sqrt{13} (- \sqrt{26})$ .

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Step 15.1.4.1

Multiply $- 1$ by $- 2$ .

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} + 2 \sqrt{13} \sqrt{26}}{143}}$

Step 15.1.4.2

Combine using the product rule for radicals.

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} + 2 \sqrt{26 \cdot 13}}{143}}$

Step 15.1.4.3

Multiply $26$ by $13$ .

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} + 2 \sqrt{338}}{143}}$

Step 15.1.5

Rewrite $338$ as $13^{2} \cdot 2$ .

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Step 15.1.5.1

Factor $169$ out of $338$ .

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} + 2 \sqrt{169 (2)}}{143}}$

Step 15.1.5.2

Rewrite $169$ as $13^{2}$ .

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} + 2 \sqrt{13^{2} \cdot 2}}{143}}$

Step 15.1.6

Pull terms out from under the radical.

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} + 2 (13 \sqrt{2})}{143}}$

Step 15.1.7

Multiply $13$ by $2$ .

$\sqrt{\frac{169 - 13 \sqrt{26} - 26 \sqrt{13} + 26 \sqrt{2}}{143}}$

Step 15.2

Cancel the common factor of $169 - 13 \sqrt{26} - 26 \sqrt{13} + 26 \sqrt{2}$ and $143$ .

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Step 15.2.1

Factor $13$ out of $169$ .

$\sqrt{\frac{13 (13) - 13 \sqrt{26} - 26 \sqrt{13} + 26 \sqrt{2}}{143}}$

Step 15.2.2

Factor $13$ out of $- 13 \sqrt{26}$ .

$\sqrt{\frac{13 (13) + 13 (- \sqrt{26}) - 26 \sqrt{13} + 26 \sqrt{2}}{143}}$

Step 15.2.3

Factor $13$ out of $13 (13) + 13 (- \sqrt{26})$ .

$\sqrt{\frac{13 (13 - \sqrt{26}) - 26 \sqrt{13} + 26 \sqrt{2}}{143}}$

Step 15.2.4

Factor $13$ out of $- 26 \sqrt{13}$ .

$\sqrt{\frac{13 (13 - \sqrt{26}) + 13 (- 2 \sqrt{13}) + 26 \sqrt{2}}{143}}$

Step 15.2.5

Factor $13$ out of $13 (13 - \sqrt{26}) + 13 (- 2 \sqrt{13})$ .

$\sqrt{\frac{13 (13 - \sqrt{26} - 2 \sqrt{13}) + 26 \sqrt{2}}{143}}$

Step 15.2.6

Factor $13$ out of $26 \sqrt{2}$ .

$\sqrt{\frac{13 (13 - \sqrt{26} - 2 \sqrt{13}) + 13 (2 \sqrt{2})}{143}}$

Step 15.2.7

Factor $13$ out of $13 (13 - \sqrt{26} - 2 \sqrt{13}) + 13 (2 \sqrt{2})$ .

$\sqrt{\frac{13 (13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2})}{143}}$

Step 15.2.8

Cancel the common factors.

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Step 15.2.8.1

Factor $13$ out of $143$ .

$\sqrt{\frac{13 (13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2})}{13 \cdot 11}}$

Step 15.2.8.2

Cancel the common factor.

$\sqrt{\frac{13 (13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2})}{13 \cdot 11}}$

Step 15.2.8.3

Rewrite the expression.

$\sqrt{\frac{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}}{11}}$

Step 16

Rewrite $\sqrt{\frac{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}}{11}}$ as $\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}}}{\sqrt{11}}$ .

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}}}{\sqrt{11}}$

Step 17

Multiply $\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 18

Combine and simplify the denominator.

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Step 18.1

Multiply $\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 18.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}}$

Step 18.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}}$

Step 18.4

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

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Step 18.5

Add $1$ and $1$ .

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{2}}$

Step 18.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 18.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

Step 18.6.2

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

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

Step 18.6.3

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

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{11^{\frac{2}{2}}}$

Step 18.6.4

Cancel the common factor of $2$ .

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Step 18.6.4.1

Cancel the common factor.

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{11^{\frac{2}{2}}}$

Step 18.6.4.2

Rewrite the expression.

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{11^{1}}$

Step 18.6.5

Evaluate the exponent.

$\frac{\sqrt{13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}} \sqrt{11}}{11}$

Step 19

Combine using the product rule for radicals.

$\frac{\sqrt{(13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}) \cdot 11}}{11}$

Step 20

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(13 - \sqrt{26} - 2 \sqrt{13} + 2 \sqrt{2}) \cdot 11}}{11}$

Decimal Form:

$0.56554921 \dots$