Surds (Cambridge IGCSE Maths): Revision notes

Section 1

What is a surd?

A surd is a root (usually a square root) that cannot be simplified to a rational number — its decimal expansion is infinite and non-repeating.

$\sqrt{4} = 2$ is NOT a surd (it simplifies to a whole number), but $\sqrt{5}$ IS a surd, because it cannot be written exactly as a fraction or terminating decimal.

Key termssurdirrational number

Section 2

How do I simplify surd expressions?

Use the rule $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ (and its reverse) to pull out square factors.

Steps:

Find the largest square number that divides into the number under the root
Split the root using that square factor
Simplify the square root of the square factor

Example: $\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$

Example (combining like surds): $\sqrt{200} - \sqrt{32} = \sqrt{100 \times 2} - \sqrt{16 \times 2} = 10\sqrt{2} - 4\sqrt{2} = 6\sqrt{2}$

Key termslike surds

Exam tip

Only surds with the SAME number under the root can be added or subtracted directly — simplify first if they don't look alike but could be.

Section 3

How do I rationalise the denominator?

Rationalising removes a surd from the denominator of a fraction.

Case 1 — single surd denominator: multiply top and bottom by that surd. $\frac{10}{\sqrt{5}} = \frac{10 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}$

Case 2 — binomial denominator: multiply top and bottom by the conjugate (same terms, opposite sign). $\frac{1}{1-\sqrt{3}} = \frac{1(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})} = \frac{1+\sqrt{3}}{1-3} = \frac{-1+\sqrt{3}}{2}$

Key termsrationaliseconjugate

Common mistake

When using the conjugate, the denominator uses the difference of two squares — (a−b)(a+b) = a² − b² — so the surd term always cancels out.

Section 4

How do I expand brackets involving surds?

Expand exactly as with algebra, using $\sqrt{a} \times \sqrt{a} = a$ .

Example: $(2 + \sqrt{3})(3 - \sqrt{3}) = 6 - 2\sqrt{3} + 3\sqrt{3} - 3 = 3 + \sqrt{3}$

Always simplify the resulting surds and collect like terms at the end.

Example

(√5 + 1)² = 5 + 2√5 + 1 = 6 + 2√5

Must Know

A surd is an unsimplifiable root — its value is irrational
$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ ; use this to extract square factors, e.g. √20 = 2√5
Only like surds (same number under the root) can be added or subtracted
Rationalise a single surd denominator by multiplying top and bottom by that surd
Rationalise a binomial surd denominator by multiplying by its conjugate
Expand surd brackets like normal algebra, remembering √a x √a = a

That's the notes covered.

Carry on to the next subtopic.