Simplify 4 square root of -1/8

$4 \sqrt{- \frac{1}{8}}$

Step 1

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

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

$4 \sqrt{i^{2} \frac{1}{8}}$

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

$4 \sqrt{i^{2} \frac{1^{2} \cdot 1}{8}}$

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

$4 \sqrt{i^{2} \frac{1^{2} \cdot 1}{2^{2} \cdot 2}}$

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

$4 \sqrt{i^{2} ({(\frac{1}{2})}^{2} \frac{1}{2})}$

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

$4 \sqrt{{(\frac{i}{2})}^{2} \frac{1}{2}}$

Step 2

Pull terms out from under the radical.

$4 (\frac{i}{2} \sqrt{\frac{1}{2}})$

Step 3

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

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

Step 4

Any root of $1$ is $1$ .

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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

$4 (\frac{i}{2} \cdot \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}})$

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

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

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

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

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

$4 (\frac{i}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

Add $1$ and $1$ .

$4 (\frac{i}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{2}})$

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

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

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

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

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

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

$4 (\frac{i}{2} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 (\frac{i}{2} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}})$

Rewrite the expression.

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

Evaluate the exponent.

$4 (\frac{i}{2} \cdot \frac{\sqrt{2}}{2})$

Step 7

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

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

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

Multiply $2$ by $2$ .

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

Step 8

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$i \sqrt{2}$

Simplify 4 square root of -1/8

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