Evaluate (-3+ square root of -27)/3

$\frac{- 3 + \sqrt{- 27}}{3}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 27$ as $- 1 (27)$ .

$\frac{- 3 + \sqrt{- 1 (27)}}{3}$

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$\frac{- 3 + \sqrt{- 1} \cdot \sqrt{27}}{3}$

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

$\frac{- 3 + i \cdot \sqrt{27}}{3}$

Step 2

Simplify the numerator.

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Step 3

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

Factor $3$ out of $- 3$ .

$\frac{3 \cdot - 1 + 3 i \sqrt{3}}{3}$

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

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

Factor $3$ out of $3 (- 1) + 3 (i \sqrt{3})$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 1 + i \sqrt{3}}{1}$

Divide $- 1 + i \sqrt{3}$ by $1$ .

$- 1 + i \sqrt{3}$