Trigonometry Examples

$\sqrt{\frac{\frac{13}{13} - \frac{5}{13}}{2}}$

Step 1

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{13 - 5}{13}}{2}}$

Step 2

Subtract

$5$ from

$13$ .

$\sqrt{\frac{\frac{8}{13}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{8}{13} \cdot \frac{1}{2}}$

Step 4

Cancel the common factor of

$2$ .

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Step 4.1

Factor

$2$ out of

$8$ .

$\sqrt{\frac{2 (4)}{13} \cdot \frac{1}{2}}$

Step 4.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 4}{13} \cdot \frac{1}{2}}$

Step 4.3

Rewrite the expression.

$\sqrt{\frac{4}{13}}$

Step 5

Rewrite

$\sqrt{\frac{4}{13}}$ as

$\frac{\sqrt{4}}{\sqrt{13}}$ .

$\frac{\sqrt{4}}{\sqrt{13}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite

$4$ as

$2^{2}$ .

$\frac{\sqrt{2^{2}}}{\sqrt{13}}$

Step 6.2

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

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

Step 7

Multiply

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

$\frac{\sqrt{13}}{\sqrt{13}}$ .

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

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply

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

$\frac{\sqrt{13}}{\sqrt{13}}$ .

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

Step 8.2

Raise

$\sqrt{13}$ to the power of

$1$ .

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

Step 8.3

Raise

$\sqrt{13}$ to the power of

$1$ .

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

Step 8.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.5

Add

$1$ and

$1$ .

$\frac{2 \sqrt{13}}{{\sqrt{13}}^{2}}$

Step 8.6

Rewrite

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

$13$ .

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Step 8.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{13}$ as

$13^{\frac{1}{2}}$ .

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

Step 8.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 8.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of

$2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Evaluate the exponent.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.55470019 \dots$