# What is the exact value of the square root of 32 over 5 the square root of 14?

Oct 6, 2015

$\frac{4 \sqrt{7}}{35}$

#### Explanation:

$\frac{\sqrt{32}}{5 \sqrt{14}}$

Simplify $\sqrt{32}$.

$\frac{\sqrt{2 \times 2 \times 2 \times 2 \times 2}}{5 \sqrt{14}} =$

$\frac{\sqrt{{2}^{2} \times {2}^{2} \times 2}}{5 \sqrt{14}} =$

Apply square root rule $\sqrt{{a}^{2}} = a$.

$\frac{2 \times 2 \sqrt{2}}{5 \sqrt{14}} =$

$\frac{4 \sqrt{2}}{5 \sqrt{14}}$

Rationalize the denominator.

$\frac{4 \sqrt{2}}{5 \sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} =$

$\frac{4 \sqrt{2} \sqrt{14}}{5 \times 14} =$

$\frac{4 \sqrt{28}}{70} =$

Simplify $\left(4 \sqrt{28}\right)$.

$\frac{4 \sqrt{2 \times 2 \times 7}}{70} =$

$\frac{4 \sqrt{{2}^{2} \times 7}}{70} =$

$\frac{4 \times 2 \sqrt{7}}{70} =$

$\frac{8 \sqrt{7}}{70}$

Simplify.

$\frac{4 \sqrt{7}}{35}$