# What is the square root of 4.5?

Sep 15, 2015

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

#### Explanation:

You know that

$4.5 = \frac{45}{10}$

This means that you can write

$\sqrt{4.5} = \sqrt{\frac{45}{10}} = \frac{\sqrt{45}}{\sqrt{10}}$

Since $45 = 3 \cdot 3 \cdot 5$, you can write

$\frac{\sqrt{45}}{\sqrt{10}} = \frac{\sqrt{3 \cdot 3 \cdot 5}}{\sqrt{10}} = \frac{3 \sqrt{5}}{\sqrt{10}}$

Rationalize the denominator by multiplying the faction by $1 = \frac{\sqrt{10}}{\sqrt{10}}$

$\frac{3 \sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{3 \cdot \sqrt{50}}{10}$

This can be further simplified to

$\frac{3}{10} \cdot \sqrt{50} = \frac{3}{10} \cdot \sqrt{{5}^{2} \cdot 2} = \frac{3}{10} \cdot 5 \cdot \sqrt{2} = \textcolor{g r e e n}{\frac{3}{2} \cdot \sqrt{2}}$