# Rationalizing the Denominator by Multiplying by a Conjugate

Rationalizing the denominator of a radical expression is a method used to eliminate radicals from a denominator. If the denominator is a binomial with a rational part and an irrational part, then you'll need to use the conjugate of the binomial.

Binomials of the form $a\sqrt{b}+c\sqrt{b}$ and $a\sqrt{b}-c\sqrt{b}$ are called conjugates. For example, $4+\sqrt{3}$ and $4-\sqrt{3}$ are conjugates.

The product of two conjugates results in a difference of two squares.

$\begin{array}{l}\left(4+\sqrt{3}\right)\left(4-\sqrt{3}\right)={4}^{2}-{\left(\sqrt{3}\right)}^{2}\\ =16-3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}13\end{array}$

Example:

Simplify.

$\frac{3}{5-\sqrt{2}}$

Multiply both the numerator and denominator by the conjugate of the denominator.

$\frac{3}{5-\sqrt{2}}=\frac{3}{5-\sqrt{2}}\cdot \frac{5+\sqrt{2}}{5+\sqrt{2}}$

The denominator is now a difference of squares .

$=\frac{3\left(5+\sqrt{2}\right)}{{5}^{2}-{\left(\sqrt{2}\right)}^{2}}$

Use the power of a product property in the denominator.

$=\frac{15+3\sqrt{2}}{25-2}$

$=\frac{15+3\sqrt{2}}{23}$