Simplify 1/( square root of -5)

$\frac{1}{\sqrt{- 5}}$

Pull out imaginary unit $i$ .

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

$\frac{1}{\sqrt{- 1 (5)}}$

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

$\frac{1}{\sqrt{- 1} \cdot \sqrt{5}}$

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

$\frac{1}{i \cdot \sqrt{5}}$

Multiply $\frac{1}{i \cdot \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{i \cdot \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Combine and simplify the denominator.

Multiply $\frac{1}{i \cdot \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5}}{i \cdot \sqrt{5} \sqrt{5}}$

Move $\sqrt{5}$ .

$\frac{\sqrt{5}}{i (\sqrt{5} \sqrt{5})}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5}}{i ({\sqrt{5}}^{1} \sqrt{5})}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5}}{i ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

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

$\frac{\sqrt{5}}{i {\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

$\frac{\sqrt{5}}{i {(5^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{5}}{i \cdot 5^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{5}}{i \cdot 5^{1}}$

Evaluate the exponent.

$\frac{\sqrt{5}}{i \cdot 5}$

Move $5$ to the left of $i$ .

$\frac{\sqrt{5}}{5 i}$