# How do you rationalize the denominator and simplify sqrt24/sqrt3?

May 13, 2016

$\frac{\sqrt{24}}{\sqrt{3}} = 2 \sqrt{2}$

#### Explanation:

We will use:

• If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$

• If $b > 0$ then $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

The normal way to solve this example is to multiply both the numerator and denominator by $\sqrt{3}$ first:

$\frac{\sqrt{24}}{\sqrt{3}} = \frac{\sqrt{24} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{72}}{3} = \frac{\sqrt{{6}^{2} \cdot 2}}{3} = \frac{\sqrt{{6}^{2}} \cdot \sqrt{2}}{3} = \frac{6 \sqrt{2}}{3} = 2 \sqrt{2}$

Alternatively, we can combine the numerator and denominator inside the square root:

$\frac{\sqrt{24}}{\sqrt{3}} = \sqrt{\frac{24}{3}} = \sqrt{8} = \sqrt{{2}^{2}} \cdot \sqrt{2} = 2 \sqrt{2}$