Simplify ( square root of 10)/( square root of 180- square root of 20)

$\frac{\sqrt{10}}{\sqrt{180} - \sqrt{20}}$

Simplify the denominator.

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$\frac{\sqrt{10}}{\sqrt{36 (5)} - \sqrt{20}}$

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{10}}{\sqrt{6^{2} \cdot 5} - \sqrt{20}}$

Pull terms out from under the radical.

$\frac{\sqrt{10}}{6 \sqrt{5} - \sqrt{20}}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\frac{\sqrt{10}}{6 \sqrt{5} - \sqrt{4 (5)}}$

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

$\frac{\sqrt{10}}{6 \sqrt{5} - \sqrt{2^{2} \cdot 5}}$

Pull terms out from under the radical.

$\frac{\sqrt{10}}{6 \sqrt{5} - (2 \sqrt{5})}$

Multiply $2$ by $- 1$ .

$\frac{\sqrt{10}}{6 \sqrt{5} - 2 \sqrt{5}}$

Subtract $2 \sqrt{5}$ from $6 \sqrt{5}$ .

$\frac{\sqrt{10}}{4 \sqrt{5}}$

Combine $\sqrt{10}$ and $\sqrt{5}$ into a single radical.

$\frac{\sqrt{\frac{10}{5}}}{4}$

Cancel the common factor of $10$ and $5$ .

Factor $5$ out of $10$ .

$\frac{\sqrt{\frac{5 \cdot 2}{5}}}{4}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{\sqrt{\frac{5 \cdot 2}{5 (1)}}}{4}$

Cancel the common factor.

$\frac{\sqrt{\frac{5 \cdot 2}{5 \cdot 1}}}{4}$

Rewrite the expression.

$\frac{\sqrt{\frac{2}{1}}}{4}$

Divide $2$ by $1$ .

$\frac{\sqrt{2}}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{4}$

Decimal Form:

$0.35355339 \dots$