Trigonometry Examples

$\sqrt{\frac{5}{10} + \frac{2 \sqrt{6}}{10}}$

Step 1

Combine the numerators over the common denominator.

$\sqrt{\frac{5 + 2 \sqrt{6}}{10}}$

Step 2

Rewrite $\sqrt{\frac{5 + 2 \sqrt{6}}{10}}$ as $\frac{\sqrt{5 + 2 \sqrt{6}}}{\sqrt{10}}$ .

$\frac{\sqrt{5 + 2 \sqrt{6}}}{\sqrt{10}}$

Step 3

Multiply $\frac{\sqrt{5 + 2 \sqrt{6}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt{5 + 2 \sqrt{6}}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{5 + 2 \sqrt{6}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.2

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

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 4.3

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

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 4.4

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

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 4.6

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

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Step 4.6.1

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

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{10^{1}}$

Step 4.6.5

Evaluate the exponent.

$\frac{\sqrt{5 + 2 \sqrt{6}} \sqrt{10}}{10}$

Step 5

Combine using the product rule for radicals.

$\frac{\sqrt{(5 + 2 \sqrt{6}) \cdot 10}}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(5 + 2 \sqrt{6}) \cdot 10}}{10}$

Decimal Form:

$0.99493615 \dots$