IGCSEMathsCambridge (CIE)ExtendedRevision NotesNumberSurdsSimplifying Surds

Simplifying Surds (Cambridge (CIE) IGCSE Maths: Extended): Revision Note

Exam code: 0580 & 0980

Written by: Amber

Reviewed by: Dan Finlay

Updated on 1 October 2026

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

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

How do I do calculations with surds?

Multiplying surds
- You can multiply numbers under square roots together
- $\sqrt{3} \times \sqrt{5} = \sqrt{3 \times 5} = \sqrt{15}$
Dividing surds
- You can divide numbers under square roots
- $\frac{\sqrt{21}}{\sqrt{7}} = \sqrt{21} \div \sqrt{7} = \sqrt{21 \div 7} = \sqrt{3}$
Factorising surds
- You can factorise numbers under square roots
- $\sqrt{35} = \sqrt{5 \times 7} = \sqrt{5} \times \sqrt{7}$
Adding or subtracting surds
- You can only add or subtract multiples of “like” surds
  - This is similar to collecting like terms when simplifying algebra
- $3 \sqrt{5} + 8 \sqrt{5} = 11 \sqrt{5}$
- $7 \sqrt{3} - 4 \sqrt{3} = 3 \sqrt{3}$
  - However $2 \sqrt{3} + 4 \sqrt{6}$ cannot be simplified
- You cannot add or subtract numbers under square roots
- Consider $\sqrt{9} + \sqrt{4} = 3 + 2 = 5$
  - This is not equal to $\sqrt{9 + 4} = \sqrt{13} = 3.60555 \dots$

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 $\sqrt{a b} = \sqrt{a} \times \sqrt{b}$ and then work out any square roots of square numbers
- E.g. $\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4 \times \sqrt{3} = 4 \sqrt{3}$

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. $\sqrt{32} + \sqrt{8}$ can be rewritten as $\sqrt{16} \sqrt{2} + \sqrt{4} \sqrt{2}$
  - This simplifies to $4 \sqrt{2} + 2 \sqrt{2}$
  - These surds can then be collected together
  - $6 \sqrt{2}$
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 ${(\sqrt{a})}^{2} = a$ can be used to simplify the expression, once expanded
- E.g. $(\sqrt{6} - 2) (\sqrt{6} + 4)$ expands to ${(\sqrt{6})}^{2} + 4 \sqrt{6} - 2 \sqrt{6} - 8$
  - This simplifies to $6 + 2 \sqrt{6} - 8$ which gives $- 2 + 2 \sqrt{6}$

Worked Example

Write $\sqrt{54} - \sqrt{24}$ in the form $\sqrt{q}$ where $q$ is a positive integer.

Answer:

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 $\sqrt{54} = \sqrt{9 \times 6} = 3 \sqrt{6}$

4 is a factor of 24, so $\sqrt{24} = \sqrt{4 \times 6} = 2 \sqrt{6}$

Simplify the whole expression by collecting the like terms

$\sqrt{54} - \sqrt{24} = 3 \sqrt{6} - 2 \sqrt{6} = \sqrt{6}$

$\sqrt{6}$

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Questions students ask about Simplifying Surds

Real questions from students revising this topic on Save My Exams

Do you always have to use the largest square factor when simplifying a surd?

No — but the largest square factor gets you to simplest form in one step; with a smaller one you have to keep going until no square factor is left.

The page's rule: if several square numbers are factors, use the largest
Your final answer must be in simplest surd form, meaning the number under the root has no square factors other than 1

Page example: √720

Largest square factor is 144: √720 = √144 × √5 = 12√5 — done in one step

Using a smaller square factor:

√720 = √4 × √180 = 2√180
180 still has a square factor (36), so carry on
2√180 = 2 × √36 × √5 = 2 × 6√5 = 12√5

Same answer, more steps — and it is easy to stop too early at 2√180, which is not fully simplified.

Tip: work down the square numbers 144, 100, 81, 64, ... from the largest that could divide in.

How do you add or subtract surds with different numbers under the root?

Simplify each surd first — if they turn into the same surd you can collect them, and if they do not, the expression cannot be simplified.

You can only add or subtract like surds, just like collecting like terms in algebra: 3√6 − 2√6 = √6
You cannot add the numbers under the roots: √9 + √4 = 5, but √13 = 3.605...

Steps

Simplify each surd separately using its largest square factor
If the numbers under the roots now match, add or subtract the numbers in front
If they still do not match, leave it as it is

Page's worked example: √54 − √24

√54 = √9 × √6 = 3√6
√24 = √4 × √6 = 2√6
3√6 − 2√6 = √6

Another: √50 − √8 = 5√2 − 2√2 = 3√2

When it will not simplify: 2√3 + 4√6 — different numbers under the roots, neither reducible.

How do you multiply surds with numbers outside the square root?

Multiply the outside numbers together, multiply the numbers under the roots together, then simplify the surd you get.

a√b × c√d = (a × c)√(b × d)