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Rationalising Denominators (Edexcel IGCSE Further Pure Maths)

Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

Rationalising Denominators

What does it mean to rationalise a denominator?

  • If a fraction has a surd in the denominator, it is often useful to rationalise it

  • 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 if the denominator is a surd?

  • STEP 1
    Multiply the top and bottom 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

    • We are multiplying by 1, so the overall value does not change

  • STEP 2
    Multiply the numerators and denominators

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

How do I rationalise the denominator if the denominator is a linear 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 in the denominator, but with the sign in the middle 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

    • We are multiplying by 1, so the overall value does not change

  • STEP 2
    Multiply out the expressions in the numerators and denominators

    • 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 minus space a square root of b space plus 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

    • Note that this is an example of 'difference of two squares'

  • 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 fraction numerator 2 open parentheses 1 space minus space square root of 3 close parentheses over denominator negative 2 end fraction space equals space minus open parentheses 1 space minus space square root of 3 close parentheses space equals space minus 1 space plus space square root of 3 space

Examiner Tips and Tricks

  • Remember that the aim is to remove the surd from the denominator

    • If this doesn't happen, check your working or rethink the expression you used in your calculation

  • Your calculator can rationalise denominators

    • You can use this to check your answer

    • But on a 'show that' question you must show your working to get full marks

Worked Example

Write fraction numerator 4 over denominator square root of 6 space minus space 2 end fraction 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 containing a surd in the denominator, so the fraction will need to be multiplied by a fraction with this expression as both the numerator and denominator, but with the sign changed

fraction numerator 4 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 4 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

When expanding the denominator, notice that it is a difference of two squares problem

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

Simplify by cancelling out the 2 in the denominator against the 4 in the numerator

2 open parentheses square root of 6 space plus space 2 close parentheses

Expand and write in the form given in the question

2 square root of 6 space plus space 4 space equals space 4 space plus space 2 square root of 6

This is now in the required form, with p equals 4, q equals 2 and r equals 6

bold 4 bold plus bold 2 square root of bold 6

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    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.