Simplifying Surds (Edexcel IGCSE Further Pure 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 $7.071067811...$

How do I calculate with surds?

•  Multiplying surds

  • You can multiply numbers under square roots together

  • $\sqrt{a}\times \sqrt{b}=\sqrt{ab}$

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

• Dividing surds

  • You can divide numbers under square roots

  • $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$

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

• Factorising surds

  • You can factorise numbers under square roots

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

  • $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$

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

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

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

    • e.g.  $3\sqrt{5}+8\sqrt{5}=11\sqrt{5}$  or  $7\sqrt{3}–4\sqrt{3}=3\sqrt{3}$

    • but  $3\sqrt{5}-4\sqrt{3}$  can't be simplified further

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

    • $\sqrt{a}+\sqrt{b}$  is not equal to  $\sqrt{a+b}$

    • $\sqrt{a}-\sqrt{b}$ is not equal to $\sqrt{a-b}$ 

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

  

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

    • e.g. $\sqrt{450}=\sqrt{9\times 50}=\sqrt{9}\times \sqrt{50}=3\times \sqrt{50}=3\sqrt{50}$

    • But 50 still has a square factor so continue

    • $3\sqrt{50}=3\sqrt{25\times 2}=3\left(\sqrt{25}\times \sqrt{2}\right)=3\left(5\times \sqrt{2}\right)=15\times \sqrt{2}=15\sqrt{2}$

• 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 $\sqrt{32}+\sqrt{8}$ by simplifying each part separately

 $\begin{array}{rcl}\sqrt{32}+\sqrt{8}& =& \left(\sqrt{16}\times \sqrt{2}\right)+\left(\sqrt{4}\times \sqrt{2}\right)\\ & =& 4\sqrt{2}+2\sqrt{2}\\ & =& 6\sqrt{2}\end{array}$

• An important skill is expanding brackets containing surds

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

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