GCSEMathsAQAHigherRevision Notes1. NumberSurdsRationalising Denominators

Rationalising Denominators (AQA GCSE Maths: Higher): Revision Note

Exam code: 8300

Written by: Amber

Reviewed by: Dan Finlay

Updated on 2 July 2026

Rationalising Denominators

What does rationalising the denominator mean?

If a fraction has a denominator containing a surd then it has an irrational denominator
- E.g. $\frac{4}{\sqrt{5}}$ or $\sqrt{\frac{2}{3}} = \frac{\sqrt{2}}{\sqrt{3}}$
The fraction can be rewritten as an equivalent fraction, but with a rational denominator
- E.g. $\frac{4 \sqrt{5}}{5}$ or $\frac{\sqrt{6}}{3}$
The numerator may contain a surd, but the denominator is rationalised

How do I rationalise denominators?

If the denominator is a surd:
- Multiply the top and bottom of the fraction by the surd on the denominator
  - $\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}$
  - This is equivalent to multiplying by 1, so does not change the value of the fraction
  - $\sqrt{b} \times \sqrt{b} = b$ so the denominator is no longer a surd
- Multiply the fractions as you would usually, and simplify if needed
  - $\frac{a \sqrt{b}}{b}$

Worked Example

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.

Answer:

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

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

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

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