Simplify ( square root of -4)^2

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

Step 1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6

Move $2$ to the left of $i$ .

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

Step 7

Apply the product rule to $2 i$ .

$2^{2} i^{2}$

Step 8

Raise $2$ to the power of $2$ .

$4 i^{2}$

Step 9

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

$4 \cdot - 1$

Step 10

Multiply $4$ by $- 1$ .

$- 4$