# Rationalizing a Binomial Denominator with Radicals

There is an unspoken law in math that a radical cannot be left in the denominator. The process of eliminating the radical from the denominator is called 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)

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 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

= answer written in equivalent a+bi form

Example 2

= multiply the numerator and denominator by the conjugate of the        denominator

=   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 denominator

Or

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

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.