Simplify square root of -81+3i

$\sqrt{- 81} + 3 i$

Simplify each term.

Rewrite $- 81$ as $- 1 (81)$ .

$\sqrt{- 1 (81)} + 3 i$

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$\sqrt{- 1} \cdot \sqrt{81} + 3 i$

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

$i \cdot \sqrt{81} + 3 i$

Rewrite $81$ as $9^{2}$ .

$i \cdot \sqrt{9^{2}} + 3 i$

Pull terms out from under the radical, assuming positive real numbers.

$i \cdot 9 + 3 i$

Move $9$ to the left of $i$ .

$9 i + 3 i$

Add $9 i$ and $3 i$ .

$12 i$