Rationalising the denominator

In a nutshell

Rationalising surds eliminates the irrational number in the denominator. You can think of this as converting to equivalent fractions with a rational denominator. Surd fractions with more complex denominators will need the conjugate surd to rationalise them.

Definitions

Rationalising	Converting a fraction with an irrational denominator to one with a rational denominator.
Conjugate	A binomial expression with the same terms as the original, except with the opposite sign in the middle.

Rationalising the denominator

Rationalising the denominator means removing the surd from the bottom of a fraction, by converting to an equivalent fraction. This is done by multiplying the numerator and the denominator by the same term.

To rationalise a surd, you multiply the numerator and the denominator by the surd in the denominator.

Example 1

Rationalise the denominator $\dfrac{1}{\sqrt7}$ .

Multiply both the numerator and the denominator by the surd in the denominator.

$\begin {aligned} \frac{1}{\sqrt7} &= \frac{1}{\sqrt7}\times {\frac{\sqrt7}{\sqrt7}}\\ &= \underline{\frac{\sqrt7}{7}} \end {aligned}$

Rationalising more complex denominators

Sometimes you can have more complex denominators like:

$\dfrac{1}{2+\sqrt2}$

Multiplying top and bottom by $\sqrt2$ will still leave a surd in the denominator. You need the conjugate expression of the denominator to eliminate the surd. In this case, the conjugate is:

$2-\sqrt2$

procedure

1.	Form the conjugate of the denominator.
2.	Multiply both the numerator and the denominator by the conjugate.
3.	Simplify the answer.

Example 2

Rationalise the denominator $\dfrac{2}{3+\sqrt3}$ .

Find the conjugate.

$3-\sqrt3$ .

Multiply both the numerator and the denominator by the conjugate.

$\begin{aligned}\begin{split} \dfrac{2}{3+\sqrt{3}} &= \dfrac{2}{(3+\sqrt{3})} \times \dfrac{(3-\sqrt{3})}{(3-\sqrt{3})} \\\newline&=\dfrac{6 - 2\sqrt{3}}{9 - 3\sqrt{3} + 3\sqrt{3} - 3} \\\newline&=\dfrac{6-2\sqrt{3}}{6} \\\newline&=\dfrac{2(3-\sqrt{3})}{6} \\\newline&=\underline{\dfrac{3-\sqrt{3}}{3}} \\\end{split}\end{aligned}$