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GCSEFurther MathsAQARevision Notes1. NumberSurdsRationalising Denominators

Rationalising Denominators (AQA GCSE Further Maths)

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Amber

Written by: Amber

Reviewed by: Dan Finlay

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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

  • To rationalise the denominator if the denominator is an expression containing a surd:
    For example fraction numerator 2 over denominator 1 space plus space square root of 3 end fraction 

    • STEP 1: Multiply the top and bottom by the expression on the denominator, but with the sign changed
      fraction numerator 2 over denominator 1 space plus space square root of 3 end fraction space cross times space fraction numerator 1 space minus space square root of 3 over denominator 1 space minus space square root of 3 end fraction

      • This ensures we are multiplying by 1; so not affecting the overall value

    • STEP 2: Multiply the expressions on the numerator and denominator together

      • open parentheses a space plus space square root of b close parentheses open parentheses a space minus space square root of b close parentheses space equals space a squared space plus space a square root of b space minus space a square root of b space minus space b space equals space a squared space minus space b so the denominator no longer contains a surd

    • STEP 3: Simplify your answer if needed
      fraction numerator 2 open parentheses 1 space minus space square root of 3 close parentheses over denominator open parentheses 1 space plus space square root of 3 close parentheses open parentheses 1 space minus space square root of 3 close parentheses end fraction space equals fraction numerator 2 open parentheses 1 space minus space square root of 3 close parentheses over denominator 1 space minus space 3 end fraction space equals space minus fraction numerator 2 open parentheses 1 space minus space square root of 3 close parentheses over denominator 2 end fraction space equals space minus open parentheses 1 space minus space square root of 3 close parentheses space equals space square root of 3 space minus space 1

 

  • To rationalise the denominator if the denominator is an expression containing more than one surd:
    For example fraction numerator 1 over denominator a square root of b space plus space c square root of d end fraction 

    • STEP 1: Multiply the top and bottom by the expression on the denominator, but with the sign changed (this is called the conjugate)
      fraction numerator 1 over denominator a square root of b plus c space square root of d end fraction space cross times space fraction numerator a square root of b minus c space square root of d over denominator a square root of b minus c space square root of d end fraction

      • This ensures we are multiplying by 1; so not affecting the overall value

    • STEP 2: Multiply the expressions on the numerator and denominator togethertable attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 space cross times space open parentheses a square root of b space minus space c square root of d close parentheses over denominator open parentheses a square root of b space plus space c square root of d close parentheses open parentheses a square root of b space minus space c square root of d close parentheses end fraction end cell equals cell space fraction numerator a square root of b space minus space c square root of d over denominator a squared open parentheses square root of b close parentheses squared space minus space a square root of b c square root of d space plus space a square root of b c square root of d minus c squared open parentheses square root of d close parentheses squared space end fraction end cell end table

      • so the denominator no longer contains a surd

    • STEP 3: Simplify your answer if needed
      fraction numerator 1 over denominator a square root of b plus c space square root of d end fraction table row blank equals blank end table table row blank blank space end table table row blank blank cell fraction numerator a square root of b space minus space c square root of d over denominator a squared b space minus c squared d end fraction end cell end table

 

Examiner Tips and Tricks

  • When you have an expression on the denominator you can use the FOIL technique from multiplying out double brackets

    • 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 the expression you are using in your calculation

Worked Example

Rationalise and simplify.

fraction numerator 4 space minus space square root of 6 over denominator square root of 6 space minus space 2 end fraction

Give your answer in the form p space plus space q square root of r where p comma space q spaceand r are integers and r has no square factors.

There is an expression on the denominator, so the fraction will need to be multiplied by a fraction with this expression on both the numerator and denominator, but with the sign changed.

fraction numerator 4 minus square root of 6 over denominator square root of 6 space minus space 2 end fraction space cross times space fraction numerator square root of 6 space plus space 2 over denominator square root of 6 space plus space 2 end fraction

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

space fraction numerator open parentheses 4 minus square root of 6 close parentheses open parentheses square root of 6 space plus space 2 close parentheses over denominator open parentheses square root of 6 space minus space 2 close parentheses open parentheses square root of 6 space plus space 2 close parentheses end fraction

Expand the numerator and the denominator. You can expand the denominator quickly by using the difference of two squares

table row blank blank cell space fraction numerator 4 square root of 6 plus 8 minus open parentheses square root of 6 close parentheses squared minus 2 square root of 6 over denominator open parentheses square root of 6 close parentheses squared space minus space 4 end fraction end cell row blank equals cell space fraction numerator 4 square root of 6 plus 8 minus 6 minus 2 square root of 6 over denominator 6 space space minus space 4 end fraction end cell end table

Simplify the numerator and denominator.

table row blank blank cell fraction numerator 2 plus 2 square root of 6 over denominator 2 end fraction end cell end table

Divide both terms in the numerator by 2.

fraction numerator 4 minus square root of 6 over denominator square root of 6 space minus space 2 end fraction space equals space 1 space plus space 1 square root of 6

bold 1 bold space bold plus bold space square root of bold 6 bold italic p bold space bold equals bold space bold 1 bold comma bold space bold italic q bold space bold equals bold space bold 1 bold comma bold space bold italic r bold space bold equals bold space bold 6

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    Previous:Simplifying SurdsNext:Counting Principles
    Author
    Reviewer
    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.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths 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.