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Simplifying Surds (AQA GCSE Maths)

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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

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
    • eg. 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
    • eg. 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
    • eg. 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
    • eg. 3 square root of 5 plus space 8 square root of 5 space equals space 11 square root of 5 space o r space 7 square root of 3 space – space 4 square root of 3 space equals space 3 square root of 3
    • Be very careful here, you can not add or subtract numbers under square roots
    • 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 Tip

  • If you are working on an exam question and your calculator gives you an answer as a surd, leave the value as a surd throughout the rest of your calculations to make sure you do not lose accuracy throughout your questions

Simplifying Surds

How do I simplify surds?

  • To simplify a surd, separate out a square factor and square root it
    • Look for the greatest square number that is a factor of the number you are simplifying
    • eg. 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

  • You can collect like terms with surds like you do with letters in algebra
  • Understanding how to simplify surds can help reduce expressions and collect 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 square root of 16 space cross times space square root of 2 space plus space square root of 4 space cross times space square root of 2 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 multiplying double brackets containing surds
    • This can be done in the same way as multiplying out double brackets algebraically and 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

Examiner Tip

  • When simplifying surds, use the fact that the one, non square factor will be the same in each part to help you find the correct, highest square factor

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

square root of bold 54 bold space bold minus bold space square root of bold 24 bold space bold equals bold space square root of bold 6

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    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.