Trigonometry Examples

$\sqrt{\frac{- \frac{1}{3}}{2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{- \frac{1}{3} \cdot \frac{1}{2}}$

Step 2

Multiply $- \frac{1}{3} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$\sqrt{- \frac{1}{2 \cdot 3}}$

Step 2.2

Multiply $2$ by $3$ .

$\sqrt{- \frac{1}{6}}$

Step 3

Rewrite $- \frac{1}{6}$ as $i^{2} \frac{1^{2}}{6}$ .

Tap for more steps...

Step 3.1

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{1}{6}}$

Step 3.2

Rewrite $1$ as $1^{2}$ .

$\sqrt{i^{2} \frac{1^{2}}{6}}$

Step 4

Pull terms out from under the radical.

$i \sqrt{\frac{1^{2}}{6}}$

Step 5

One to any power is one.

$i \sqrt{\frac{1}{6}}$

Step 6

Rewrite $\sqrt{\frac{1}{6}}$ as $\frac{\sqrt{1}}{\sqrt{6}}$ .

$i \frac{\sqrt{1}}{\sqrt{6}}$

Step 7

Any root of $1$ is $1$ .

$i \frac{1}{\sqrt{6}}$

Step 8

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$i (\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 9

Combine and simplify the denominator.

Tap for more steps...

Step 9.1

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$i \frac{\sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 9.2

Raise $\sqrt{6}$ to the power of $1$ .

$i \frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 9.3

Raise $\sqrt{6}$ to the power of $1$ .

$i \frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 9.4

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

$i \frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$i \frac{\sqrt{6}}{{\sqrt{6}}^{2}}$

Step 9.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$i \frac{\sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$i \frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$i \frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.6.4.1

Cancel the common factor.

$i \frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$i \frac{\sqrt{6}}{6^{1}}$

Step 9.6.5

Evaluate the exponent.

$i \frac{\sqrt{6}}{6}$

Step 10

Combine $i$ and $\frac{\sqrt{6}}{6}$ .

$\frac{i \sqrt{6}}{6}$