Simplify ( 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 $i$ .

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

Rewrite $- 3$ as $- 1 (3)$ .

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

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

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

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

Step 2

Cancel the common factor of $i$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

Step 4

Divide $81$ by $3$ .

$\sqrt{27}$

Step 5

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

Factor $9$ out of $27$ .

$\sqrt{9 (3)}$

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

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

Step 6

Pull terms out from under the radical.

$3 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{3}$

Decimal Form:

$5.19615242 \dots$