Trigonometry Examples

$\sqrt{\frac{1 + \frac{5}{- 13}}{2}}$

Step 1

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

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

Step 2

To write $\frac{5}{- 13}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\sqrt{\frac{\frac{13}{13} + \frac{5}{- 13} \cdot \frac{- 1}{- 1}}{2}}$

Step 3

Write each expression with a common denominator of $13$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{- 13}$ by $\frac{- 1}{- 1}$ .

$\sqrt{\frac{\frac{13}{13} + \frac{5 \cdot - 1}{- 13 \cdot - 1}}{2}}$

Step 3.2

Multiply $- 13$ by $- 1$ .

$\sqrt{\frac{\frac{13}{13} + \frac{5 \cdot - 1}{13}}{2}}$

Step 4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{13 + 5 \cdot - 1}{13}}{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $5$ by $- 1$ .

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

Step 5.2

Subtract $5$ from $13$ .

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $8$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

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

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

Step 9

Simplify the numerator.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

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

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

Step 11

Combine and simplify the denominator.

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Step 11.1

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

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

Step 11.2

Raise $\sqrt{13}$ to the power of $1$ .

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

Step 11.3

Raise $\sqrt{13}$ to the power of $1$ .

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

Step 11.4

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

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

Step 11.5

Add $1$ and $1$ .

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

Step 11.6

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

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Step 11.6.1

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 11.6.2

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

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

Step 11.6.3

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

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

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

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

Step 11.6.4.2

Rewrite the expression.

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

Step 11.6.5

Evaluate the exponent.

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.55470019 \dots$