Simplify ( square root of -7)/( square root of 63)

$\frac{\sqrt{- 7}}{\sqrt{63}}$

Pull out imaginary unit $i$ .

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{\sqrt{- 1 (7)}}{\sqrt{63}}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$\frac{\sqrt{- 1} \cdot \sqrt{7}}{\sqrt{63}}$

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

$\frac{i \cdot \sqrt{7}}{\sqrt{63}}$

Combine $\sqrt{7}$ and $\sqrt{63}$ into a single radical.

$i \cdot \sqrt{\frac{7}{63}}$

Cancel the common factor of $7$ and $63$ .

Factor $7$ out of $7$ .

$i \cdot \sqrt{\frac{7 (1)}{63}}$

Cancel the common factors.

Factor $7$ out of $63$ .

$i \cdot \sqrt{\frac{7 \cdot 1}{7 \cdot 9}}$

Cancel the common factor.

$i \cdot \sqrt{\frac{7 \cdot 1}{7 \cdot 9}}$

Rewrite the expression.

$i \cdot \sqrt{\frac{1}{9}}$

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$i \cdot \frac{\sqrt{1}}{\sqrt{9}}$

Any root of $1$ is $1$ .

$i \cdot \frac{1}{\sqrt{9}}$

Simplify the denominator.

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

$i \cdot \frac{1}{\sqrt{3^{2}}}$

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

$i \cdot \frac{1}{3}$

Combine $i$ and $\frac{1}{3}$ .

$\frac{i}{3}$