Trigonometry Examples

$\sqrt{\frac{1 - (\frac{- 2 \sqrt{5}}{5})}{2}}$

Step 1

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Multiply $5$ by $2$ .

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

Step 7

Rewrite $\sqrt{\frac{5 + 2 \sqrt{5}}{10}}$ as $\frac{\sqrt{5 + 2 \sqrt{5}}}{\sqrt{10}}$ .

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

Step 8

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

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

Step 9

Combine and simplify the denominator.

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Step 9.1

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

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

Step 9.2

Raise $\sqrt{10}$ to the power of $1$ .

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

Step 9.3

Raise $\sqrt{10}$ to the power of $1$ .

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

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

Step 9.6.2

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

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

Step 9.6.3

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

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

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

Step 9.6.5

Evaluate the exponent.

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

Step 10

Combine using the product rule for radicals.

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.97324898 \dots$