# How do you simplify (square root 2) + 2 (square root 2) + (square root 8) / (square root 3)?

Sep 6, 2015

$\frac{5}{3} \cdot \sqrt{6}$

#### Explanation:

I'll assume that the expression looks like this

$\frac{\sqrt{2} + 2 \sqrt{2} + \sqrt{8}}{\sqrt{3}}$

Start by focusing on the numerator. More specifically, notice that you can rewrite $\sqrt{8}$ as

$\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \cdot \sqrt{2}$

The numerator will then take the form

$\sqrt{2} + 2 \sqrt{2} + 2 \sqrt{2} = 5 \sqrt{2}$

Next, you need to rationalize the denominator. To do that, multiply the fraction by $1 = \frac{\sqrt{3}}{\sqrt{3}}$

$\frac{5 \sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5 \cdot \sqrt{2} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{5 \cdot \sqrt{2 \cdot 3}}{\sqrt{3 \cdot 3}}$

The final form of the expression will be

$\frac{5 \cdot \sqrt{6}}{\sqrt{{3}^{2}}} = \textcolor{g r e e n}{\frac{5}{3} \cdot \sqrt{6}}$