# Rationalizing a Binomial Denominator with Radicals

*rationalizing*. When the denominator is a binomial (two terms) the

*conjugate*of the denominator has to be used to rationalize.

Let's start be reviewing

**.**

*conjugate*
$3{+}\sqrt{2}$ is a binomial with a radical

$3{-}\sqrt{2}$ the conjugate (change the sign in the middle)

$3{-}\sqrt{2}$ the conjugate (change the sign in the middle)

**Example 1**

- $\frac{4}{\sqrt{5}-3}$

= $\frac{4}{\left(\sqrt{5}-3\right)}.\frac{\left(\sqrt{5}+3\right)}{\left(\sqrt{5}+3\right)}$ multiply the numerator and denominator by the

= $\frac{4\sqrt{5}+12}{5+3\sqrt{5}-3\sqrt{5}-9}$ use the distributive property to simplify the top and bottom

= $\frac{4\sqrt{5}+12}{-4}$ combine like terms and notice that by multiplying by the

= $\frac{4\sqrt{5}}{-4}+\frac{12}{-4}$ prepare to reduce fractions

= $-\sqrt{5}-3$ reduce fractions

Or

= $-3-\sqrt{5}$ answer written in equivalent

Example 2*conjugate*of the denominator= $\frac{4\sqrt{5}+12}{5+3\sqrt{5}-3\sqrt{5}-9}$ use the distributive property to simplify the top and bottom

= $\frac{4\sqrt{5}+12}{-4}$ combine like terms and notice that by multiplying by the

*conjugate*that radicals are eliminated in the denominator= $\frac{4\sqrt{5}}{-4}+\frac{12}{-4}$ prepare to reduce fractions

= $-\sqrt{5}-3$ reduce fractions

Or

= $-3-\sqrt{5}$ answer written in equivalent

*a+bi*form- $\frac{2+2}{3-\sqrt{2}}$

= $\frac{\left(2+\sqrt{2}\right)}{\left(3-\sqrt{2}\right)}.\frac{\left(3+\sqrt{2}\right)}{\left(3+\sqrt{2}\right)}$ multiply the numerator and denominator by the

= $\frac{6+2\sqrt{2}+3\sqrt{2}+2}{9+3\sqrt{2}-3\sqrt{2}-2}$ use the distributive property to simplify the top and bottom

= $\frac{8+5\sqrt{2}}{7}$ combine like terms and notice that by multiplying by the

Or

= $\frac{8}{7}+\frac{5\sqrt{2}}{7}$ answer written in equivalent

To rationalize a radical expression, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial is obtained by changing the middle sign to its opposite.*conjugate*of the denominator= $\frac{6+2\sqrt{2}+3\sqrt{2}+2}{9+3\sqrt{2}-3\sqrt{2}-2}$ use the distributive property to simplify the top and bottom

= $\frac{8+5\sqrt{2}}{7}$ combine like terms and notice that by multiplying by the

*conjugate*that radicals are eliminated in the denominatorOr

= $\frac{8}{7}+\frac{5\sqrt{2}}{7}$ answer written in equivalent

*a+bi*formTo link to this **Rationalizing a Binomial Denominator with Radicals** page, copy the following code to your site: