Surds — An Examiner's Guide

Surds are the topic that most often makes a Year 11 student say "why are we doing this?" The answer is simple: an exam question that asks for an "exact answer" or a "simplified form" wants a surd, not a decimal. If you write 1.41 instead of √2 you'll lose accuracy marks even when the answer is right. This guide walks through what surds are, the four operations you need, and the routine that catches out the most students — rationalising the denominator.

What a surd is — and why we use them

A surd is the square root of a non-square positive integer that can't be simplified to a whole number. So √2, √3 and √5 are surds. √4 is not — it equals 2 exactly. √9 is not — it equals 3.

We use surds because they are exact. √2 is exactly √2, whereas 1.4142… is an approximation that goes on forever. In GCSE Higher questions that say "give your answer in the form a + b√c" or "give an exact answer", you must work in surds throughout.

The two key rules

Everything below is built from those two rules.

Simplifying surds

The trick to simplifying a surd is to spot a square factor inside the square root.

Adding and subtracting surds

You can only add or subtract like surds — surds with the same number under the root. Treat them like algebraic terms.

If the surds aren't immediately alike, simplify each one first — they often turn out to be alike after simplification.

Multiplying surds

For multiplication, multiply the numbers outside the roots and the numbers inside the roots separately.

Rationalising the denominator

This is the topic students struggle with most. "Rationalising" means getting rid of the surd from the bottom of a fraction — because a fraction with a rational denominator is considered fully simplified.

Single-term denominator

Multiply top and bottom by the surd in the denominator.

Two-term denominator (Higher only)

For something like 1 / (3 + √2), multiply top and bottom by the conjugate — the same expression with the sign flipped (so 3 − √2). This works because (a + b)(a − b) = a² − b² eliminates the surd.

The mistakes that cost the most marks

Exam-day checklist for surds

1. Read the question. Does it want an "exact answer" or "form a + b√c"? If yes, work in surds throughout — don't decimalise.
2. For simplifying, look for the biggest square factor (4, 9, 16, 25, 36, 49, 64, 81, 100).
3. For adding/subtracting, simplify each surd first, then combine like terms.
4. For a single-term denominator, multiply top and bottom by that surd.
5. For a two-term denominator, multiply by the conjugate.