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Surds

A surd is a root that cannot be simplified to a whole number, like $2$ . Leaving answers as surds keeps them exact. This is a Higher-tier topic that rewards careful, tidy working.

40 min Video by Maths Genie AQA GCSE Maths

SurdsWatch the walkthrough, then read the notes below.

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What you'll learn

Simplify surds
Multiply and divide surds
Add and subtract like surds
Rationalise the denominator

Simplifying surds

Look for a factor that is a perfect square. Since $ab = a b$ , you can pull the square root of the square factor outside: $12 = 4 \times 3 = 23$ .

ab = a b

Split off a perfect-square factor.

50 = 25 \times 2

, and 25 is a perfect square.

50 = 252 = 52

Answer

52

Rationalising the denominator

It is untidy to leave a surd on the bottom of a fraction. Multiply top and bottom by that surd to clear it: $\frac{1}{2} = \frac{2}{2}$ .

Tip — You can only add or subtract “like” surds: $23 + 43 = 63$ , but $3 + 2$ will not simplify.

Remember these

a \times a = a

A surd times itself removes the root.

\frac{1}{a} = \frac{a}{a}

Rationalising the denominator.

Watch out for these

Writing

a + b = a + b

You cannot add surds inside one root; only combine like surds.

Leaving a surd in the denominator.

Rationalise it by multiplying top and bottom by the surd.

Key takeaways

Simplify by splitting off a perfect-square factor.
Add/subtract only like surds.
Rationalise to clear a surd from the denominator.

Test yourself

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