Rationalising Denominators

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If a fraction has a surd on the denominator, it is not in its simplest form and must be rationalised
Rationalising a denominator changes a fraction with surds in the denominator into an equivalent fraction
- The denominator will be an integer and any surds are in the numerator

Remember that the aim is to remove the surd from the denominator, so if this doesn't happen you need to check your working or rethink your working
If a question involving rationalising a denominator appears in a calculator paper, you can use your calculator to check your answer by typing in the un-rationalised fraction - you will still need to show full working though if asked!

Write $\frac{4}{\sqrt{6}}$ in the form $q \sqrt{r}$ where $q$ is a fraction in its simplest form and $r$ has no square factors.

There is a surd on the denominator, so the fraction will need to be multiplied by a fraction with this surd on both the numerator and denominator

$\frac{4}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}$

Multiply the fractions together by multiplying across the numerator and the denominator.

$\frac{4 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}$

By multiplying out the denominator, you will notice that the surds are removed

$\begin{array}{rcl} \frac{4 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}} & = & \frac{4 \sqrt{6}}{6} \end{array}$

Rewriting in the form $q \sqrt{r}$ and simplifying the fraction

$\frac{4 \sqrt{6}}{6} = \frac{4}{6} \times \sqrt{6} = \frac{2}{3} \sqrt{6}$

$\frac{2}{3} \sqrt{6} q = \frac{2}{3} r = 6$

Rationalising Denominators (AQA GCSE Maths)