Simplifying Surds (AQA GCSE Further Maths): Revision Note

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. $5\sqrt{2}$  rather than $7.071067812$

How do I calculate with surds?

•  Multiplying surds

  • You can multiply numbers under square roots together

  • eg. $\sqrt{3}\times \sqrt{5}=\sqrt{3\times 5}=\sqrt{15}$

• Dividing surds

  • You can divide numbers under square roots

  • eg. $\sqrt{21}÷\sqrt{7}=\sqrt{21÷7}=\sqrt{3}$

• Factorising surds

  • You can factorise numbers under square roots

  • eg. $\sqrt{35}=\sqrt{5\times 7}=\sqrt{5}\times \sqrt{7}$

• Adding or subtracting surds is very like adding or subtracting letters in algebra – you can only add or subtract multiples of “like” surds

  • eg. $3\sqrt{5}+8\sqrt{5}=11\sqrt{5}or7\sqrt{3}–4\sqrt{3}=3\sqrt{3}$

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

  • Think about $\sqrt{9}+\sqrt{4}=3+2=5$ 

    • It is not equal to $\sqrt{9+4}=\sqrt{13}=3.60555\dots$

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. $\sqrt{48}=\sqrt{16\times 3}=\sqrt{16}\times \sqrt{3}=4\times \sqrt{3}=4\sqrt{3}$

• 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 $\sqrt{32}+\sqrt{8}$ by simplifying each part separately
     $\begin{array}{rcl}\sqrt{32}+\sqrt{8}& =& \sqrt{16}\times \sqrt{2}+\sqrt{4}\times \sqrt{2}\\ & =& 4\sqrt{2}+2\sqrt{2}\\ & =& 6\sqrt{2}\end{array}$

• 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 ${\left(\sqrt{a}\right)}^2=a$ can be used to simplify the expression, once expanded