Trigonometry Examples

$\sqrt{\frac{1 + \frac{1}{5 \sqrt{2}}}{2}}$

Step 1

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

$\sqrt{\frac{\frac{5 \sqrt{2}}{5 \sqrt{2}} + \frac{1}{5 \sqrt{2}}}{2}}$

Step 2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{5 \sqrt{2} + 1}{5 \sqrt{2}}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{5 \sqrt{2} + 1}{5 \sqrt{2}} \cdot \frac{1}{2}}$

Step 4

Multiply $\frac{5 \sqrt{2} + 1}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{5 \sqrt{2} + 1}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \frac{1}{2}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{5 \sqrt{2} + 1}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 \sqrt{2} \sqrt{2}} \cdot \frac{1}{2}}$

Step 5.2

Move $\sqrt{2}$ .

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 (\sqrt{2} \sqrt{2})} \cdot \frac{1}{2}}$

Step 5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 ({\sqrt{2}}^{1} \sqrt{2})} \cdot \frac{1}{2}}$

Step 5.4

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})} \cdot \frac{1}{2}}$

Step 5.5

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

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}} \cdot \frac{1}{2}}$

Step 5.6

Add $1$ and $1$ .

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 {\sqrt{2}}^{2}} \cdot \frac{1}{2}}$

Step 5.7

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

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Step 5.7.1

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

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}} \cdot \frac{1}{2}}$

Step 5.7.2

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

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{2}}$

Step 5.7.3

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

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}} \cdot \frac{1}{2}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}} \cdot \frac{1}{2}}$

Step 5.7.4.2

Rewrite the expression.

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 \cdot 2^{1}} \cdot \frac{1}{2}}$

Step 5.7.5

Evaluate the exponent.

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{5 \cdot 2} \cdot \frac{1}{2}}$

Step 6

Multiply $5$ by $2$ .

$\sqrt{\frac{(5 \sqrt{2} + 1) \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 7

Apply the distributive property.

$\sqrt{\frac{5 \sqrt{2} \sqrt{2} + 1 \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 8

Multiply $5 \sqrt{2} \sqrt{2}$ .

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Step 8.1

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{5 ({\sqrt{2}}^{1} \sqrt{2}) + 1 \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 8.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 1 \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 8.3

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

$\sqrt{\frac{5 {\sqrt{2}}^{1 + 1} + 1 \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 8.4

Add $1$ and $1$ .

$\sqrt{\frac{5 {\sqrt{2}}^{2} + 1 \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 9

Multiply $\sqrt{2}$ by $1$ .

$\sqrt{\frac{5 {\sqrt{2}}^{2} + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

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

$\sqrt{\frac{5 {(2^{\frac{1}{2}})}^{2} + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 10.1.2

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

$\sqrt{\frac{5 \cdot 2^{\frac{1}{2} \cdot 2} + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 10.1.3

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

$\sqrt{\frac{5 \cdot 2^{\frac{2}{2}} + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 10.1.4

Cancel the common factor of $2$ .

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Step 10.1.4.1

Cancel the common factor.

$\sqrt{\frac{5 \cdot 2^{\frac{2}{2}} + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 10.1.4.2

Rewrite the expression.

$\sqrt{\frac{5 \cdot 2^{1} + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 10.1.5

Evaluate the exponent.

$\sqrt{\frac{5 \cdot 2 + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 10.2

Multiply $5$ by $2$ .

$\sqrt{\frac{10 + \sqrt{2}}{10} \cdot \frac{1}{2}}$

Step 11

Multiply $\frac{10 + \sqrt{2}}{10} \cdot \frac{1}{2}$ .

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Step 11.1

Multiply $\frac{10 + \sqrt{2}}{10}$ by $\frac{1}{2}$ .

$\sqrt{\frac{10 + \sqrt{2}}{10 \cdot 2}}$

Step 11.2

Multiply $10$ by $2$ .

$\sqrt{\frac{10 + \sqrt{2}}{20}}$

Step 12

Rewrite $\sqrt{\frac{10 + \sqrt{2}}{20}}$ as $\frac{\sqrt{10 + \sqrt{2}}}{\sqrt{20}}$ .

$\frac{\sqrt{10 + \sqrt{2}}}{\sqrt{20}}$

Step 13

Simplify the denominator.

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Step 13.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 13.1.1

Factor $4$ out of $20$ .

$\frac{\sqrt{10 + \sqrt{2}}}{\sqrt{4 (5)}}$

Step 13.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{10 + \sqrt{2}}}{\sqrt{2^{2} \cdot 5}}$

Step 13.2

Pull terms out from under the radical.

$\frac{\sqrt{10 + \sqrt{2}}}{2 \sqrt{5}}$

Step 14

Multiply $\frac{\sqrt{10 + \sqrt{2}}}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{10 + \sqrt{2}}}{2 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 15

Combine and simplify the denominator.

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Step 15.1

Multiply $\frac{\sqrt{10 + \sqrt{2}}}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 \sqrt{5} \sqrt{5}}$

Step 15.2

Move $\sqrt{5}$ .

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 (\sqrt{5} \sqrt{5})}$

Step 15.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 15.4

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 15.5

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

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}$

Step 15.6

Add $1$ and $1$ .

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 {\sqrt{5}}^{2}}$

Step 15.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 15.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 {(5^{\frac{1}{2}})}^{2}}$

Step 15.7.2

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

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 15.7.3

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

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 15.7.4

Cancel the common factor of $2$ .

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Step 15.7.4.1

Cancel the common factor.

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 15.7.4.2

Rewrite the expression.

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 \cdot 5^{1}}$

Step 15.7.5

Evaluate the exponent.

$\frac{\sqrt{10 + \sqrt{2}} \sqrt{5}}{2 \cdot 5}$

Step 16

Combine using the product rule for radicals.

$\frac{\sqrt{(10 + \sqrt{2}) \cdot 5}}{2 \cdot 5}$

Step 17

Multiply $2$ by $5$ .

$\frac{\sqrt{(10 + \sqrt{2}) \cdot 5}}{10}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(10 + \sqrt{2}) \cdot 5}}{10}$

Decimal Form:

$0.75545395 \dots$