Simplify ( square root of -12)^2

${(\sqrt{- 12})}^{2}$

Step 1

Rewrite $- 12$ as $- 1 (12)$ .

${\sqrt{- 1 (12)}}^{2}$

Step 2

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

${(\sqrt{- 1} \cdot \sqrt{12})}^{2}$

Step 3

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

${(i \cdot \sqrt{12})}^{2}$

Step 4

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

${(i \cdot \sqrt{4 (3)})}^{2}$

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

${(i \cdot \sqrt{2^{2} \cdot 3})}^{2}$

Step 5

Pull terms out from under the radical.

${(i \cdot (2 \sqrt{3}))}^{2}$

Step 6

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

${(2 \cdot i \sqrt{3})}^{2}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 i \sqrt{3}$ .

${(2 i)}^{2} {\sqrt{3}}^{2}$

Apply the product rule to $2 i$ .

$2^{2} i^{2} {\sqrt{3}}^{2}$

Step 8

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

$4 i^{2} {\sqrt{3}}^{2}$

Step 9

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

$4 \cdot - 1 {\sqrt{3}}^{2}$

Step 10

Multiply $4$ by $- 1$ .

$- 4 {\sqrt{3}}^{2}$

Step 11

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 4 {(3^{\frac{1}{2}})}^{2}$

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

$- 4 \cdot 3^{\frac{1}{2} \cdot 2}$

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

$- 4 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 4 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$- 4 \cdot 3^{1}$

Evaluate the exponent.

$- 4 \cdot 3$

Step 12

Multiply $- 4$ by $3$ .

$- 12$

Simplify ( square root of -12)^2

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