Rationalising Denominators (Edexcel IGCSE Further Pure Maths): Revision Note

Rationalising Denominators

What does it mean to rationalise a denominator?

• If a fraction has a surd in the denominator, it is often useful to rationalise it

• 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

How do I rationalise the denominator if the denominator is a surd?

• STEP 1
  Multiply the top and bottom by the surd in the denominator:
  $\frac{a}{\sqrt{b}}= \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}$
  

  • We are multiplying by 1, so the overall value does not change

• STEP 2
  Multiply the numerators and denominators

$\sqrt{b}\times \sqrt{b}=b$ so the denominator is no longer a surd

• STEP 3
  Simplify your answer if needed

How do I rationalise the denominator if the denominator is a linear expression containing a surd?

For example $\frac{2}{1+\sqrt{3}}$ 

• STEP 1
  Multiply the top and bottom by the expression in the denominator, but with the sign in the middle changed
  $\frac{2}{1+\sqrt{3}}\times \frac{1-\sqrt{3}}{1-\sqrt{3}}$
  

  • We are multiplying by 1, so the overall value does not change

• STEP 2
  Multiply out the expressions in the numerators and denominators

  • $\left(a+\sqrt{b}\right)\left(a-\sqrt{b}\right)=a^2-a\sqrt{b}+a\sqrt{b}-b=a^2-b$ so the denominator no longer contains a surd

  • Note that this is an example of 'difference of two squares'

• STEP 3
  Simplify your answer if needed

$\frac{2\left(1-\sqrt{3}\right)}{\left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right)}=\frac{2\left(1-\sqrt{3}\right)}{1-3}=\frac{2\left(1-\sqrt{3}\right)}{-2}=-\left(1-\sqrt{3}\right)=-1+\sqrt{3}$