Algebra Examples

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

Step 1

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

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Step 1.1

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

$\sqrt{i^{2} \frac{1}{3}}$

Step 1.2

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

$\sqrt{i^{2} \frac{1^{2}}{3}}$

Step 2

Pull terms out from under the radical.

$i \sqrt{\frac{1^{2}}{3}}$

Step 3

One to any power is one.

$i \sqrt{\frac{1}{3}}$

Step 4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$i \frac{\sqrt{1}}{\sqrt{3}}$

Step 5

Any root of $1$ is $1$ .

$i \frac{1}{\sqrt{3}}$

Step 6

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

$i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

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

$i \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2

Raise $\sqrt{3}$ to the power of $1$ .

$i \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.3

Raise $\sqrt{3}$ to the power of $1$ .

$i \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.4

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

$i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$i \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$i \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$i \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$i \frac{\sqrt{3}}{3^{1}}$

Step 7.6.5

Evaluate the exponent.

$i \frac{\sqrt{3}}{3}$

Step 8

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$\frac{i \sqrt{3}}{3}$