National 5MathsSQARevision NotesNumerical SkillsSurdsRationalising Denominators

Rationalising Denominators (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

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 a denominator?

  • If the denominator is a surd:

    • Multiply the top and bottom of the fraction by the surd in the denominator

      • fraction numerator a over denominator square root of b end fraction equals blank fraction numerator a over denominator square root of b end fraction blank cross times blank fraction numerator square root of b over denominator square root of 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

Examiner Tips and Tricks

Questions about rationalising denominators or simplifying surds will almost always appear on the non-calculator paper in the exam.

Worked Example

Express fraction numerator 8 over denominator square root of 14 end fraction​ with a rational denominator. Give your answer in its simplest form.

Answer:

Multiply the top and bottom of the fraction by the surd in the denominator

fraction numerator 8 over denominator square root of 14 end fraction equals fraction numerator 8 over denominator square root of 14 end fraction cross times fraction numerator square root of 14 over denominator square root of 14 end fraction

Multiply the fractions as you would usually

equals space fraction numerator 8 square root of 14 over denominator square root of 14 space cross times square root of 14 end fraction

square root of a cross times square root of a equals a, so

equals fraction numerator 8 square root of 14 over denominator 14 end fraction

Simplify by dividing the top and bottom by 2

fraction numerator 4 square root of 14 over denominator 7 end fraction

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.