Simplify (-8+ square root of -20)/24

$\frac{- 8 + \sqrt{- 20}}{24}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{- 8 + \sqrt{- 1 (20)}}{24}$

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$\frac{- 8 + \sqrt{- 1} \cdot \sqrt{20}}{24}$

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

$\frac{- 8 + i \cdot \sqrt{20}}{24}$

Step 2

Simplify the numerator.

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\frac{- 8 + i \cdot \sqrt{4 (5)}}{24}$

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

$\frac{- 8 + i \cdot \sqrt{2^{2} \cdot 5}}{24}$

Pull terms out from under the radical.

$\frac{- 8 + i \cdot (2 \sqrt{5})}{24}$

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

$\frac{- 8 + 2 i \sqrt{5}}{24}$

Step 3

Simplify terms.

Cancel the common factor of $- 8 + 2 i \sqrt{5}$ and $24$ .

Factor $2$ out of $- 8$ .

$\frac{2 \cdot - 4 + 2 i \sqrt{5}}{24}$

Factor $2$ out of $2 i \sqrt{5}$ .

$\frac{2 \cdot - 4 + 2 (i \sqrt{5})}{24}$

Factor $2$ out of $2 \cdot - 4 + 2 (i \sqrt{5})$ .

$\frac{2 \cdot (- 4 + i \sqrt{5})}{24}$

Cancel the common factors.

Factor $2$ out of $24$ .

$\frac{2 \cdot (- 4 + i \sqrt{5})}{2 (12)}$

Cancel the common factor.

$\frac{2 \cdot (- 4 + i \sqrt{5})}{2 \cdot 12}$

Rewrite the expression.

$\frac{- 4 + i \sqrt{5}}{12}$

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

$\frac{- 1 (4) + i \sqrt{5}}{12}$

Factor $- 1$ out of $i \sqrt{5}$ .

$\frac{- 1 (4) - (- i \sqrt{5})}{12}$

Factor $- 1$ out of $- 1 (4) - (- i \sqrt{5})$ .

$\frac{- 1 (4 - i \sqrt{5})}{12}$

Move the negative in front of the fraction.

$- \frac{4 - i \sqrt{5}}{12}$