Simplify (-6+ square root of -54)/3

$\frac{- 6 + \sqrt{- 54}}{3}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 54$ as $- 1 (54)$ .

$\frac{- 6 + \sqrt{- 1 (54)}}{3}$

Rewrite $\sqrt{- 1 (54)}$ as $\sqrt{- 1} \cdot \sqrt{54}$ .

$\frac{- 6 + \sqrt{- 1} \cdot \sqrt{54}}{3}$

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

$\frac{- 6 + i \cdot \sqrt{54}}{3}$

Step 2

Simplify the numerator.

Rewrite $54$ as $3^{2} \cdot 6$ .

Factor $9$ out of $54$ .

$\frac{- 6 + i \cdot \sqrt{9 (6)}}{3}$

Rewrite $9$ as $3^{2}$ .

$\frac{- 6 + i \cdot \sqrt{3^{2} \cdot 6}}{3}$

Pull terms out from under the radical.

$\frac{- 6 + i \cdot (3 \sqrt{6})}{3}$

Move $3$ to the left of $i$ .

$\frac{- 6 + 3 i \sqrt{6}}{3}$

Step 3

Cancel the common factor of $- 6 + 3 i \sqrt{6}$ and $3$ .

Factor $3$ out of $- 6$ .

$\frac{3 \cdot - 2 + 3 i \sqrt{6}}{3}$

Factor $3$ out of $3 i \sqrt{6}$ .

$\frac{3 \cdot - 2 + 3 (i \sqrt{6})}{3}$

Factor $3$ out of $3 (- 2) + 3 (i \sqrt{6})$ .

$\frac{3 (- 2 + i \sqrt{6})}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$\frac{3 (- 2 + i \sqrt{6})}{3 (1)}$

Cancel the common factor.

$\frac{3 (- 2 + i \sqrt{6})}{3 \cdot 1}$

Rewrite the expression.

$\frac{- 2 + i \sqrt{6}}{1}$

Divide $- 2 + i \sqrt{6}$ by $1$ .

$- 2 + i \sqrt{6}$