Simplifying Surds (Edexcel IGCSE Maths A) : Revision Note

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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 7.071067812...

Examples of surds and not-surds

How do I do calculations with surds?

  •  Multiplying surds

    • You can multiply numbers under square roots together

    • 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 21 over denominator square root of 7 end fraction equals 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

    • 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

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

      • This is similar to collecting like terms when simplifying algebra

    • 3 square root of 5 plus space 8 square root of 5 space equals space 11 square root of 5 space

    • 7 square root of 3 space – space 4 square root of 3 space equals space 3 square root of 3

      • However 2 square root of 3 plus 4 square root of 6 cannot be simplified

    • You cannot add or subtract numbers under square roots

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

      • This 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 an answer as a surd, leave the value as a surd throughout the rest of your working.

This will ensure you do not lose accuracy throughout your working.

Simplifying Surds

How do I simplify surds?

  • To simplify a surd, factorise the number using a square number, if possible

    • If multiple square numbers are a factor, use the largest

  • Use the fact that square root of a b end root equals square root of a cross times square root of b and then work out any square roots of square numbers

    • 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

Simplifying root 8 to 2 root 2 and root 720 to 12 root 5
  • When simplifying multiple surds, simplify each separately

    • This may produce surds which can then be collected together

      • E.g. square root of 32 space plus space square root of 8 can be rewritten as square root of 16 square root of 2 space plus space square root of 4 square root of 2

      • This simplifies to 4 square root of 2 plus 2 square root of 2

      • These surds can then be collected together

      • 6 square root of 2

  • You may have to expand double brackets containing surds

    • This can be done in the same way as multiplying out double brackets algebraically, and then simplifying

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

    • E.g. open parentheses square root of 6 minus 2 close parentheses open parentheses square root of 6 plus 4 close parentheses expands to open parentheses square root of 6 close parentheses squared space plus 4 square root of 6 minus 2 square root of 6 minus 8

      • This simplifies to 6 plus 2 square root of 6 minus 8 which gives negative 2 plus 2 square root of 6

Worked Example

Write square root of 54 space minus space square root of 24 in the form square root of q where q is a positive integer.

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

square root of 6

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Amber

Author: Amber

Expertise: Maths Content Creator

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

Reviewer: 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.

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