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Rationalising Denominators (AQA GCSE Maths)

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Amber

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Amber

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AQA Maths: HigherEdexcel Maths: Higher

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. fraction numerator 4 over denominator square root of 5 end fraction or square root of 2 over 3 end root equals fraction numerator square root of 2 over denominator square root of 3 end fraction

  • The fraction can be rewritten as an equivalent fraction, but with a rational denominator

    • E.g. fraction numerator 4 square root of 5 over denominator 5 end fraction or fraction numerator square root of 6 over denominator 3 end fraction

  • 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

      • 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 is equivalent to multiplying by 1, so does not change the value of the fraction

      • square root of b space cross times space square root of b space equals space b so the denominator is no longer a surd

    • Multiply the fractions as you would usually, and simplify if needed

      • fraction numerator a square root of b over denominator b end fraction

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 2 over 3 square root of 6

bold 2 over bold 3 square root of bold 6 bold italic q bold space bold equals bold 2 over bold 3 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.