Evaluate square root of 1/2 square root of -8

$\sqrt{\frac{1}{2}} \sqrt{- 8}$

Step 1

Rewrite $- 8$ as $- 1 (8)$ .

$\sqrt{\frac{1}{2}} \sqrt{- 1 (8)}$

Step 2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\sqrt{\frac{1}{2}} (\sqrt{- 1} \cdot \sqrt{8})$

Step 3

Rewrite $\sqrt{- 1}$ as $i$ .

$\sqrt{\frac{1}{2}} (i \cdot \sqrt{8})$

Step 4

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

$\frac{\sqrt{1}}{\sqrt{2}} (i \cdot \sqrt{8})$

Step 5

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{2}} (i \cdot \sqrt{8})$

Step 6

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

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} (i \cdot \sqrt{8})$

Step 7

Combine and simplify the denominator.

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

$\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} (i \cdot \sqrt{8})$

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{2}} (i \cdot \sqrt{8})$

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

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

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

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

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

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

$\frac{\sqrt{2}}{2^{\frac{2}{2}}} (i \cdot \sqrt{8})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{2}}{2^{\frac{2}{2}}} (i \cdot \sqrt{8})$

Rewrite the expression.

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

Evaluate the exponent.

$\frac{\sqrt{2}}{2} (i \cdot \sqrt{8})$

Step 8

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

Cancel the common factor of $2$ .

Factor $2$ out of $i \cdot (2 \sqrt{2})$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Add $1$ and $1$ .

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

Step 15

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$2^{1} i$

Evaluate the exponent.

$2 i$

Evaluate square root of 1/2 square root of -8

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