Simplify the Radical Expression ( square root of -81)/( square root of 3)

$\frac{\sqrt{- 81}}{\sqrt{3}}$

Step 1

Pull out imaginary unit $i$ .

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

$\frac{\sqrt{- 1 (81)}}{\sqrt{3}}$

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

$\frac{\sqrt{- 1} \cdot \sqrt{81}}{\sqrt{3}}$

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

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

Step 2

Combine $\sqrt{81}$ and $\sqrt{3}$ into a single radical.

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

Step 3

Divide $81$ by $3$ .

$i \cdot \sqrt{27}$

Step 4

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

Factor $9$ out of $27$ .

$i \cdot \sqrt{9 (3)}$

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

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

Step 5

Pull terms out from under the radical.

$i \cdot (| 3 | \sqrt{3})$

Step 6

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$i \cdot (3 \sqrt{3})$

Step 7

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

$3 i \sqrt{3}$