# How do we rationalize a denominator?

Apr 3, 2016

Multiply the whole fraction by the denominator over itself.

#### Explanation:

If you have a fraction where the denominator (bottom number) is irrational, such as a square root, then multiply the whole fraction by the denominator divided by itself.

Take the example

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

which has a rational denominator.

Multiply the entire fraction by the denom. divided by itself, like so

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

More examples are

$\frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{3 \sqrt{7}}{7}$

$\frac{5}{\sqrt{18}} = \frac{5}{3 \sqrt{2}}$
$\frac{5}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5 \sqrt{2}}{6}$

In this last example I turned $\sqrt{18}$ into $3 \sqrt{2}$ by laws of surds, and then, because $3$ is already rational, I only multiplied by $\frac{\sqrt{2}}{\sqrt{2}}$, not $\frac{3 \sqrt{2}}{3 \sqrt{2}}$.