Evaluate (-12+ square root of -28)/32

$\frac{- 12 + \sqrt{- 28}}{32}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 28$ as $- 1 (28)$ .

$\frac{- 12 + \sqrt{- 1 (28)}}{32}$

Rewrite $\sqrt{- 1 (28)}$ as $\sqrt{- 1} \cdot \sqrt{28}$ .

$\frac{- 12 + \sqrt{- 1} \cdot \sqrt{28}}{32}$

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

$\frac{- 12 + i \cdot \sqrt{28}}{32}$

Step 2

Simplify the numerator.

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$\frac{- 12 + i \cdot \sqrt{4 (7)}}{32}$

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

$\frac{- 12 + i \cdot \sqrt{2^{2} \cdot 7}}{32}$

Pull terms out from under the radical.

$\frac{- 12 + i \cdot (2 \sqrt{7})}{32}$

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

$\frac{- 12 + 2 i \sqrt{7}}{32}$

Step 3

Simplify terms.

Cancel the common factor of $- 12 + 2 i \sqrt{7}$ and $32$ .

Factor $2$ out of $- 12$ .

$\frac{2 \cdot - 6 + 2 i \sqrt{7}}{32}$

Factor $2$ out of $2 i \sqrt{7}$ .

$\frac{2 \cdot - 6 + 2 (i \sqrt{7})}{32}$

Factor $2$ out of $2 \cdot - 6 + 2 (i \sqrt{7})$ .

$\frac{2 \cdot (- 6 + i \sqrt{7})}{32}$

Cancel the common factors.

Factor $2$ out of $32$ .

$\frac{2 \cdot (- 6 + i \sqrt{7})}{2 (16)}$

Cancel the common factor.

$\frac{2 \cdot (- 6 + i \sqrt{7})}{2 \cdot 16}$

Rewrite the expression.

$\frac{- 6 + i \sqrt{7}}{16}$

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- 1 (6) + i \sqrt{7}}{16}$

Factor $- 1$ out of $i \sqrt{7}$ .

$\frac{- 1 (6) - (- i \sqrt{7})}{16}$

Factor $- 1$ out of $- 1 (6) - (- i \sqrt{7})$ .

$\frac{- 1 (6 - i \sqrt{7})}{16}$

Move the negative in front of the fraction.

$- \frac{6 - i \sqrt{7}}{16}$