# What is the simplest radical form of (4sqrt(90)) /( 3sqrt(18))?

Apr 6, 2018

$\frac{4}{3} \sqrt{2}$

#### Explanation:

We should simplify each root individually.

$\sqrt{90} = \sqrt{9 \cdot 10}$

Recall that $\sqrt{a \cdot b} = \sqrt{a} \sqrt{b} ,$ so

$\sqrt{9 \cdot 10} = \sqrt{3} \sqrt{10} = 3 \sqrt{10}$

Now,

$\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \sqrt{2} = 3 \sqrt{2}$

Thus, we have

$\frac{4 \left(3\right) \sqrt{10}}{3 \left(3\right) \sqrt{2}} = \frac{12 \sqrt{10}}{9 \sqrt{2}}$

Recalling that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} , \frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}} = \sqrt{5}$

Moreover, $\frac{12}{9} = \frac{4}{3.}$

So, the simplest form is

$\frac{4}{3} \sqrt{2}$