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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.

$\begin{array}{l} (4 + \sqrt{3}) (4 - \sqrt{3}) = 4^{2} - {(\sqrt{3})}^{2} \\ = 16 - 3 or 13 \end{array}$

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 .

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

Use the power of a product property in the denominator.

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

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