Simplify ( square root of -5)^2

${(\sqrt{- 5})}^{2}$

Step 1

Rewrite $- 5$ as $- 1 (5)$ .

${\sqrt{- 1 (5)}}^{2}$

Step 2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

${(\sqrt{- 1} \cdot \sqrt{5})}^{2}$

Step 3

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

${(i \cdot \sqrt{5})}^{2}$

Step 4

Apply the product rule to $i \sqrt{5}$ .

$i^{2} {\sqrt{5}}^{2}$

Step 5

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

$- 1 {\sqrt{5}}^{2}$

Step 6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$- 1 {(5^{\frac{1}{2}})}^{2}$

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

$- 1 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$- 1 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 1 \cdot 5^{\frac{2}{2}}$

Rewrite the expression.

$- 1 \cdot 5^{1}$

Evaluate the exponent.

$- 1 \cdot 5$

Step 7

Multiply $- 1$ by $5$ .

$- 5$