Rationalising Denominators

Surds & Exact Values

Rationalising Denominators

What does it mean to rationalise a denominator?

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

How do I rationalise the denominator of a surd?

To rationalise the denominator if the denominator is a surd
- STEP 1: Multiply the top and bottom by the surd on the denominator :
  - $fraction numerator a over denominator square root of straight b end fraction equals blank fraction numerator a over denominator square root of straight b end fraction blank cross times blank fraction numerator square root of straight b over denominator square root of straight b end fraction$
  - This ensures we are multiplying by 1; so not affecting the overall value
- STEP 2: Multiply the numerator and denominators together
  - $square root of b space cross times space square root of b space equals space b$ so the denominator is no longer a surd
- STEP 3: Simplify your answer if needed

Exam Tip

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!

Worked example

Write $fraction numerator 4 over denominator square root of 6 space end fraction$ in the form $space q square root of r$ where $q space$ 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

$fraction numerator 4 over denominator square root of 6 space end fraction space cross times space fraction numerator square root of 6 space over denominator square root of 6 space end fraction$

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

$space fraction numerator 4 cross times square root of 6 over denominator square root of 6 cross times square root of 6 end fraction$

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

$table row cell space fraction numerator 4 cross times square root of 6 over denominator square root of 6 cross times square root of 6 end fraction end cell equals cell space fraction numerator 4 square root of 6 over denominator 6 end fraction end cell end table$

Rewriting in the form $q square root of r$ and simplifying the fraction

$fraction numerator 4 square root of 6 over denominator 6 end fraction equals 4 over 6 cross times square root of 6 space equals fraction numerator space 2 over denominator 3 end fraction square root of 6$

$fraction numerator bold 2 bold space over denominator bold 3 end fraction square root of bold 6 bold italic q bold space bold equals fraction numerator bold space bold 2 over denominator bold 3 end fraction bold italic r bold space bold equals bold space bold 6$