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GCSEMathsAQARevision Notes1. NumberSurdsRationalising Denominators

Rationalising Denominators (AQA GCSE Maths)

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Rationalising Denominators

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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

 

Examiner 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 spaceis 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

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.