Trigonometry Examples

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

Step 1

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $5$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\sqrt{(5 + \sqrt{11}) \frac{1}{5 - \sqrt{11}}}$

Step 7

Multiply $\frac{1}{5 - \sqrt{11}}$ by $\frac{5 + \sqrt{11}}{5 + \sqrt{11}}$ .

$\sqrt{(5 + \sqrt{11}) (\frac{1}{5 - \sqrt{11}} \cdot \frac{5 + \sqrt{11}}{5 + \sqrt{11}})}$

Step 8

Simplify terms.

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Step 8.1

Multiply $\frac{1}{5 - \sqrt{11}}$ by $\frac{5 + \sqrt{11}}{5 + \sqrt{11}}$ .

$\sqrt{(5 + \sqrt{11}) \frac{5 + \sqrt{11}}{(5 - \sqrt{11}) (5 + \sqrt{11})}}$

Step 8.2

Expand the denominator using the FOIL method.

$\sqrt{(5 + \sqrt{11}) \frac{5 + \sqrt{11}}{25 + 5 \sqrt{11} - 5 \sqrt{11} - {\sqrt{11}}^{2}}}$

Step 8.3

Simplify.

$\sqrt{(5 + \sqrt{11}) \frac{5 + \sqrt{11}}{14}}$

Step 8.4

Apply the distributive property.

$\sqrt{5 \frac{5 + \sqrt{11}}{14} + \sqrt{11} \frac{5 + \sqrt{11}}{14}}$

Step 8.5

Combine $5$ and $\frac{5 + \sqrt{11}}{14}$ .

$\sqrt{\frac{5 (5 + \sqrt{11})}{14} + \sqrt{11} \frac{5 + \sqrt{11}}{14}}$

Step 8.6

Combine $\sqrt{11}$ and $\frac{5 + \sqrt{11}}{14}$ .

$\sqrt{\frac{5 (5 + \sqrt{11})}{14} + \frac{\sqrt{11} (5 + \sqrt{11})}{14}}$

Step 8.7

Combine the numerators over the common denominator.

$\sqrt{\frac{5 (5 + \sqrt{11}) + \sqrt{11} (5 + \sqrt{11})}{14}}$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\sqrt{\frac{5 \cdot 5 + 5 \sqrt{11} + \sqrt{11} (5 + \sqrt{11})}{14}}$

Step 9.2

Multiply $5$ by $5$ .

$\sqrt{\frac{25 + 5 \sqrt{11} + \sqrt{11} (5 + \sqrt{11})}{14}}$

Step 9.3

Apply the distributive property.

$\sqrt{\frac{25 + 5 \sqrt{11} + \sqrt{11} \cdot 5 + \sqrt{11} \sqrt{11}}{14}}$

Step 9.4

Move $5$ to the left of $\sqrt{11}$ .

$\sqrt{\frac{25 + 5 \sqrt{11} + 5 \cdot \sqrt{11} + \sqrt{11} \sqrt{11}}{14}}$

Step 9.5

Combine using the product rule for radicals.

$\sqrt{\frac{25 + 5 \sqrt{11} + 5 \cdot \sqrt{11} + \sqrt{11 \cdot 11}}{14}}$

Step 9.6

Simplify each term.

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Step 9.6.1

Multiply $11$ by $11$ .

$\sqrt{\frac{25 + 5 \sqrt{11} + 5 \sqrt{11} + \sqrt{121}}{14}}$

Step 9.6.2

Rewrite $121$ as $11^{2}$ .

$\sqrt{\frac{25 + 5 \sqrt{11} + 5 \sqrt{11} + \sqrt{11^{2}}}{14}}$

Step 9.6.3

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

$\sqrt{\frac{25 + 5 \sqrt{11} + 5 \sqrt{11} + 11}{14}}$

Step 10

Simplify terms.

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Step 10.1

Add $25$ and $11$ .

$\sqrt{\frac{36 + 5 \sqrt{11} + 5 \sqrt{11}}{14}}$

Step 10.2

Add $5 \sqrt{11}$ and $5 \sqrt{11}$ .

$\sqrt{\frac{36 + 10 \sqrt{11}}{14}}$

Step 10.3

Cancel the common factor of $36 + 10 \sqrt{11}$ and $14$ .

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Step 10.3.1

Factor $2$ out of $36$ .

$\sqrt{\frac{2 (18) + 10 \sqrt{11}}{14}}$

Step 10.3.2

Factor $2$ out of $10 \sqrt{11}$ .

$\sqrt{\frac{2 (18) + 2 (5 \sqrt{11})}{14}}$

Step 10.3.3

Factor $2$ out of $2 (18) + 2 (5 \sqrt{11})$ .

$\sqrt{\frac{2 (18 + 5 \sqrt{11})}{14}}$

Step 10.3.4

Cancel the common factors.

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Step 10.3.4.1

Factor $2$ out of $14$ .

$\sqrt{\frac{2 (18 + 5 \sqrt{11})}{2 \cdot 7}}$

Step 10.3.4.2

Cancel the common factor.

$\sqrt{\frac{2 (18 + 5 \sqrt{11})}{2 \cdot 7}}$

Step 10.3.4.3

Rewrite the expression.

$\sqrt{\frac{18 + 5 \sqrt{11}}{7}}$

Step 11

Rewrite $\sqrt{\frac{18 + 5 \sqrt{11}}{7}}$ as $\frac{\sqrt{18 + 5 \sqrt{11}}}{\sqrt{7}}$ .

$\frac{\sqrt{18 + 5 \sqrt{11}}}{\sqrt{7}}$

Step 12

Multiply $\frac{\sqrt{18 + 5 \sqrt{11}}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{18 + 5 \sqrt{11}}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply $\frac{\sqrt{18 + 5 \sqrt{11}}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 13.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 13.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 13.4

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

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 13.5

Add $1$ and $1$ .

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 13.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 13.6.2

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

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 13.6.3

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

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 13.6.4

Cancel the common factor of $2$ .

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Step 13.6.4.1

Cancel the common factor.

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 13.6.4.2

Rewrite the expression.

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{7^{1}}$

Step 13.6.5

Evaluate the exponent.

$\frac{\sqrt{18 + 5 \sqrt{11}} \sqrt{7}}{7}$

Step 14

Combine using the product rule for radicals.

$\frac{\sqrt{(18 + 5 \sqrt{11}) \cdot 7}}{7}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(18 + 5 \sqrt{11}) \cdot 7}}{7}$

Decimal Form:

$2.22271146 \dots$