Simplify ( square root of -32)( square root of -2)

$(\sqrt{- 32}) (\sqrt{- 2})$

Step 1

Rewrite $- 32$ as $- 1 (32)$ .

$\sqrt{- 1 (32)} \sqrt{- 2}$

Step 2

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$\sqrt{- 1} \cdot \sqrt{32} \sqrt{- 2}$

Step 3

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

$i \cdot \sqrt{32} \sqrt{- 2}$

Step 4

Rewrite $- 2$ as $- 1 (2)$ .

$i \cdot \sqrt{32} \sqrt{- 1 (2)}$

Step 5

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$i \cdot \sqrt{32} (\sqrt{- 1} \cdot \sqrt{2})$

Step 6

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

$i \cdot \sqrt{32} (i \cdot \sqrt{2})$

Step 7

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$i \cdot \sqrt{16 (2)} (i \cdot \sqrt{2})$

Rewrite $16$ as $4^{2}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

Move $4$ to the left of $i$ .

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

Step 10

Multiply $4 i \sqrt{2} (i \sqrt{2})$ .

Raise $i$ to the power of $1$ .

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

Raise $i$ to the power of $1$ .

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

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

$4 i^{1 + 1} \sqrt{2} \sqrt{2}$

Add $1$ and $1$ .

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

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

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

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

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

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

$4 i^{2} {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

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

Step 11

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

$4 \cdot - 1 {\sqrt{2}}^{2}$

Step 12

Multiply $4$ by $- 1$ .

$- 4 {\sqrt{2}}^{2}$

Step 13

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

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

$- 4 {(2^{\frac{1}{2}})}^{2}$

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

$- 4 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$- 4 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 4 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$- 4 \cdot 2^{1}$

Evaluate the exponent.

$- 4 \cdot 2$

Step 14

Multiply $- 4$ by $2$ .

$- 8$

Simplify ( square root of -32)( square root of -2)

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