GCSEMathsWJECMaths & Numeracy (Double Award)HigherRevision NotesNumberSurdsSimplifying Fractions with Surds

Simplifying Fractions with Surds (WJEC GCSE Maths & Numeracy (Double Award)): Revision Note

Exam code: 3320

Jamie Wood

Written by: Jamie Wood

Reviewed by: Mark Curtis

Updated on

WJEC Maths & Numeracy (Double Award): HigherAQA Maths: Higher

Simplifying Fractions with Surds

How do I simplify fractions containing surds?

  • 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 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

  • An alternate method is to rewrite the numerator in terms of the denominator, then cancel common factors

    • E.g. the numerator of fraction numerator 6 over denominator square root of 2 end fraction can be rewritten in terms of square root of 2

      • fraction numerator 6 over denominator square root of 2 end fraction equals fraction numerator 3 cross times 2 over denominator square root of 2 end fraction equals fraction numerator 3 square root of 2 square root of 2 over denominator square root of 2 end fraction

      • because 2 equals square root of 2 cross times square root of 2

    • Then cancel out the common factors on the top and bottom

      • fraction numerator 6 over denominator square root of 2 end fraction equals fraction numerator 3 square root of 2 up diagonal strike square root of 2 end strike over denominator up diagonal strike square root of 2 end strike end fraction equals 3 square root of 2

    • This method will not work for all cases

      • E.g. to simplify fraction numerator 3 over denominator square root of 2 end fraction you would need to multiply by fraction numerator square root of 2 over denominator square root of 2 end fraction

  • You may have to simplify the numerator first, often by expanding

    • See the worked example below

Worked Example

Simplify fraction numerator open parentheses 6 minus square root of 3 close parentheses open parentheses 6 plus square root of 3 close parentheses over denominator square root of 3 end fraction.

Write your answer in the form a square root of b where a and b are integers.

Answer:

Expand the numerator

fraction numerator 36 minus 6 square root of 3 plus 6 square root of 3 minus square root of 3 square root of 3 over denominator square root of 3 end fraction

The middle terms cancel and square root of 3 cross times square root of 3 equals 3

table row blank equals cell fraction numerator 36 minus 3 over denominator square root of 3 end fraction end cell row blank equals cell fraction numerator 33 over denominator square root of 3 end fraction end cell end table

Rewrite this so that it does not have a surd in the denominator

There are two methods for this

Method 1

Multiply the numerator and denominator by fraction numerator square root of 3 over denominator square root of 3 end fraction

fraction numerator 33 over denominator square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction

Recall that square root of 3 cross times square root of 3 equals 3 on the denominator

fraction numerator 33 square root of 3 over denominator 3 end fraction

Simplify using 33 over 3 equals 11

11 square root of 3

Method 2

fraction numerator 33 over denominator square root of 3 end fraction

First write the numerator 33 in terms of 3

33 equals 3 cross times 11

Then write 3 in terms of square root of 3 (using that 3 equals square root of 3 cross times square root of 3)

33 equals square root of 3 cross times square root of 3 cross times 11

Substitute this back into the numerator

fraction numerator square root of 3 cross times square root of 3 cross times 11 over denominator square root of 3 end fraction

Cancel out the common factors

fraction numerator square root of 3 cross times up diagonal strike square root of 3 end strike cross times 11 over denominator up diagonal strike square root of 3 end strike end fraction

11 square root of 3

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.