Find the Exact Value ( square root of 2)/( square root of 72+ square root of 50- square root of 8)

$\frac{\sqrt{2}}{\sqrt{72} + \sqrt{50} - \sqrt{8}}$

Step 1

Simplify the denominator.

Rewrite $72$ as $6^{2} \cdot 2$ .

Factor $36$ out of $72$ .

$\frac{\sqrt{2}}{\sqrt{36 (2)} + \sqrt{50} - \sqrt{8}}$

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

$\frac{\sqrt{2}}{\sqrt{6^{2} \cdot 2} + \sqrt{50} - \sqrt{8}}$

Pull terms out from under the radical.

$\frac{\sqrt{2}}{6 \sqrt{2} + \sqrt{50} - \sqrt{8}}$

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$\frac{\sqrt{2}}{6 \sqrt{2} + \sqrt{25 (2)} - \sqrt{8}}$

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{2}}{6 \sqrt{2} + \sqrt{5^{2} \cdot 2} - \sqrt{8}}$

Pull terms out from under the radical.

$\frac{\sqrt{2}}{6 \sqrt{2} + 5 \sqrt{2} - \sqrt{8}}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\frac{\sqrt{2}}{6 \sqrt{2} + 5 \sqrt{2} - \sqrt{4 (2)}}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Add $6 \sqrt{2}$ and $5 \sqrt{2}$ .

$\frac{\sqrt{2}}{11 \sqrt{2} - 2 \sqrt{2}}$

Subtract $2 \sqrt{2}$ from $11 \sqrt{2}$ .

$\frac{\sqrt{2}}{9 \sqrt{2}}$

Step 2

Cancel the common factor of $\sqrt{2}$ .

Cancel the common factor.

$\frac{\sqrt{2}}{9 \sqrt{2}}$

Rewrite the expression.

$\frac{1}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{9}$

Decimal Form:

$0.\overline{1}$