Rationalise the denominator

Topic summary

Rationalising the denominator means rewriting a fraction so that the denominator does not contain any surds (irrational numbers). This process makes the fraction easier to work with, particularly in further algebraic operations.

Rationalising with a Single Surd

If the denominator contains a single surd, multiply both the numerator and the denominator by that surd.

\[\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\]

Rationalising with a Binomial Surd (Difference of Squares)

When the denominator is in the form \(a + \sqrt{b}\) or \(a – \sqrt{b}\)​, multiply the numerator and denominator by the conjugate. The conjugate is the same binomial, but with the opposite sign between the terms.

\[\frac{1}{2 + \sqrt{5}} \times \frac{2 – \sqrt{5}}{2 – \sqrt{5}} = \frac{2 – \sqrt{5}}{(2 + \sqrt{5})(2 – \sqrt{5})}\]

Use the difference of squares formula:

\[(2 + \sqrt{5})(2 – \sqrt{5}) = 2^2 – (\sqrt{5})^2 = 4 – 5 = -1\]

So:

\[\frac{2 – \sqrt{5}}{2 + \sqrt{5}} = \frac{2 – \sqrt{5}}{-1} = – (2 – \sqrt{5}) = -2 + \sqrt{5}\]