Algebra Examples

$\sqrt{- \frac{1}{8}}$

Step 1

Rewrite $- \frac{1}{8}$ as ${(\frac{i}{2})}^{2} \frac{1}{2}$ .

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Step 1.1

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

$\sqrt{i^{2} \frac{1}{8}}$

Step 1.2

Factor the perfect power $1^{2}$ out of $1$ .

$\sqrt{i^{2} \frac{1^{2} \cdot 1}{8}}$

Step 1.3

Factor the perfect power $2^{2}$ out of $8$ .

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

Step 1.4

Rearrange the fraction $\frac{1^{2} \cdot 1}{2^{2} \cdot 2}$ .

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

Step 1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 4

Any root of $1$ is $1$ .

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

Step 5

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

$\frac{i}{2} (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 6.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

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

Step 6.6.5

Evaluate the exponent.

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

Step 7

Multiply $\frac{i}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $2$ .

$\frac{i \sqrt{2}}{4}$