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Simplifying Surds (Edexcel IGCSE Further Pure Maths)

Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

Surds & Exact Values

What is a surd?

  • A surd is the square root of a non-square integer

  • Using surds lets you leave answers in exact form

    • e.g. 5 square root of 2  rather than 7.071067811...

Surd and not surd, A Level & AS Level Pure Maths Revision Notes

How do I calculate with surds?

  •  Multiplying surds

    • You can multiply numbers under square roots together

    • square root of a cross times square root of b equals square root of a b end root

      • e.g. square root of 3 space cross times space square root of 5 space equals square root of space 3 cross times 5 space end root equals space square root of 15

  • Dividing surds

    • You can divide numbers under square roots

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

      • e.g. square root of 21 space divided by space square root of 7 equals space square root of 21 space divided by space 7 space end root equals space square root of 3

  • Factorising surds

    • You can factorise numbers under square roots

      • This lets you split a single square root into a product of square roots

    • square root of a b end root equals square root of a cross times square root of b

      • e.g. square root of 35 space equals square root of 5 space cross times space 7 space end root equals space square root of 5 space cross times square root of 7

  • Adding or subtracting surds is very like adding or subtracting letters in algebra

    • you can only add or subtract multiples of “like” surds

      • e.g.  3 square root of 5 plus space 8 square root of 5 space equals space 11 square root of 5  or  7 square root of 3 space – space 4 square root of 3 space equals space 3 square root of 3

      • but  3 square root of 5 minus 4 square root of 3  can't be simplified further

    • Be very careful here, you cannot add or subtract numbers under square roots

      • square root of a plus square root of b  is not equal to  square root of a plus b end root

      • square root of a minus square root of b is not equal to square root of a minus b end root 

    • Think about square root of 9 space end root plus space square root of 4 equals space 3 space plus space 2 space equals space 5 

      • It is not equal to square root of 9 plus 4 end root space equals space square root of 13 space equals space 3.60555 horizontal ellipsis

Examiner Tips and Tricks

  • If your calculator gives you an answer as a surd

    • Leave the value as a surd throughout the rest of your calculations

    • This will make sure you do not lose accuracy

    • Round only at the very end (if necessary)

  • A question might ask for an 'exact value' answer

    • In that case leave your answer as a surd

Simplifying Surds

How do I simplify surds?

  • To simplify a surd, separate out any square factors and take their square root

    • Look for the greatest square number that is a factor of the number you are simplifying

      • e.g. square root of 48 space equals space square root of 16 space cross times space 3 end root space equals space square root of 16 space cross times space square root of 3 equals space 4 space cross times space square root of 3 space equals space 4 square root of 3

    2-1-2-surds-simplify

    • If you don't spot the greatest square factor the first time continue the process

      • e.g. square root of 450 equals square root of 9 cross times 50 end root equals square root of 9 cross times square root of 50 equals 3 cross times square root of 50 equals 3 square root of 50

      • But 50 still has a square factor so continue

      • 3 square root of 50 equals 3 square root of 25 cross times 2 end root equals 3 open parentheses square root of 25 cross times square root of 2 close parentheses equals 3 open parentheses 5 cross times square root of 2 close parentheses equals 15 cross times square root of 2 equals 15 square root of 2

  • You can collect like terms with surds like you do with letters in algebra

  • Understanding how to simplify surds can help with simplifying expressions and collecting like terms

    • e.g. simplify square root of 32 space plus space square root of 8 by simplifying each part separately

 table row cell square root of 32 space plus space square root of 8 space end cell equals cell space open parentheses square root of 16 space cross times space square root of 2 close parentheses space plus space open parentheses square root of 4 space cross times space square root of 2 close parentheses space end cell row blank equals cell space 4 square root of 2 space plus space 2 square root of 2 space end cell row blank equals cell space 6 square root of 2 end cell end table

  • An important skill is expanding brackets containing surds

    • This is done in the same way as expanding brackets algebraically

    • But the property  open parentheses square root of a close parentheses squared equals a  can be used to simplify the expression, once expanded

Examiner Tips and Tricks

  • In exam questions different surds being simplified will often have the same non-square factor

    • This can help you find the correct highest square factors

    • e.g. square root of 48 plus square root of 75 can be rewritten as square root of 3 cross times 16 end root space plus space square root of 3 cross times 25 end root

      • This then simplifies easily to 4 square root of 3 plus 5 square root of 3 equals 9 square root of 3

Worked Example

Write square root of 54 space minus space square root of 24 in the form p square root of q where p and q are integers and q has no square factors.

Simplify both surds separately by finding the highest square number that is a factor of each of them

9 is a factor of 54, so 

square root of 54 space equals space square root of 9 space cross times space 6 end root space equals space 3 square root of 6

4 is a factor of 24, so

square root of 24 space equals space square root of 4 space cross times space 6 end root space equals space 2 square root of 6

Simplify the whole expression by collecting the like terms

 square root of 54 space minus square root of 24 space equals space 3 square root of 6 space minus 2 square root of 6 space space equals space square root of 6


This is in the required form with  p equals 1  and  q equals 6

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.