Simplifying Surds | Cambridge O Level Maths Revision Notes 2025

Surds & Exact Values

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 $7.071067812$

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

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. $\sqrt{21} \div \sqrt{7} = \sqrt{21 \div 7} = \sqrt{3}$
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. $3 \sqrt{5} + 8 \sqrt{5} = 11 \sqrt{5} o r 7 \sqrt{3} - 4 \sqrt{3} = 3 \sqrt{3}$
- 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 $\sqrt{9 + 4} = \sqrt{13} = 3.60555 \dots$

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