SurdsEdexcel GCSE Maths: Revision notes

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

• √2, √3, √5 are surds
• √4 = 2 and √9 = 3 are NOT surds, because they simplify to whole numbers

Calculating exactly with surds (and with multiples of π) means leaving an answer in surd form (e.g. 3√2) rather than converting to a rounded decimal, which keeps the answer precise.

To simplify a surd, find the largest square number factor and split it out.

√(a × b) = √a × √b

• √12 = √4 × √3 = √4 × √3 = 2√3
• √50 = √25 × √2 = 5√2
• √72 = √36 × √2 = 6√2

Always check for the largest square factor first, otherwise the surd may not be fully simplified.

Adding/subtracting surds: only 'like surds' (the same number under the root) can be combined, similar to collecting like terms.

• 3√2 + 5√2 = 8√2
• 3√2 + 2√3 cannot be simplified further (unlike surds)

Multiplying surds: √a × √b = √(ab), and expressions can be expanded using the same rules as algebra.

• √3 × √12 = √36 = 6
• (√a + √b)² = a + 2√(ab) + b (expand as you would (x + y)²)

Surd expressions involving squares should always be expanded and then simplified.

Rationalising the denominator means rewriting a fraction so that the denominator no longer contains a surd, by multiplying top and bottom by a suitable expression.

Form 1/√a: multiply top and bottom by √a.

• 1/√5 = (1 × √5)/(√5 × √5) = √5/5

Form (a + b√c)/(d + e√f): multiply top and bottom by the conjugate of the denominator (same terms, opposite sign in the middle), which uses the difference of two squares to remove the surd.

• 1/(2 + √3): multiply by (2 − √3)/(2 − √3) → (2 − √3)/(4 − 3) = 2 − √3