Simplifying Surds

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 $7.071067811 . . .$

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
- - e.g. $\sqrt{3} \times \sqrt{5} = \sqrt{3 \times 5} = \sqrt{15}$
Dividing surds
- You can divide numbers under square roots
- - e.g. $\sqrt{21} \div \sqrt{7} = \sqrt{21 \div 7} = \sqrt{3}$
Factorising surds
- You can factorise numbers under square roots
  - This lets you split a single square root into a product of square roots
- - 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 $\sqrt{9 + 4} = \sqrt{13} = 3.60555 \dots$

Exam Tip

If your calculator gives you an answer as a surd
- Leave the value as a surd throughout the rest of your calculations
- This will make sure you do not lose accuracy
- Round only at the very end (if necessary)
A question might ask for an 'exact value' answer
- In that case leave your answer as a surd

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

2-1-2-surds-simplify

- 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
  - $3 \sqrt{50} = 3 \sqrt{25 \times 2} = 3 (\sqrt{25} \times \sqrt{2}) = 3 (5 \times \sqrt{2}) = 15 \times \sqrt{2} = 15 \sqrt{2}$
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

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

An important skill is expanding brackets containing surds
- This is done in the same way as expanding brackets algebraically
- But the property ${(\sqrt{a})}^{2} = a$ can be used to simplify the expression, once expanded

Exam Tip

In exam questions different surds being simplified will often have the same non-square factor
- This can help you find the correct highest square factors
- e.g. can be rewritten as
  - This then simplifies easily to $4 \sqrt{3} + 5 \sqrt{3} = 9 \sqrt{3}$

Worked example

Write $\sqrt{54} - \sqrt{24}$ in the form $p \sqrt{q}$ 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

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

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

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

This is in the required form with $p = 1$ and $q = 6$

$\sqrt{6}$

Simplifying Surds (Edexcel IGCSE Further Maths)

Revision Note

Surds & Exact Values

What is a surd?

How do I calculate with surds?

Exam Tip