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Surds appear on every GCSE Higher paper and often separate a grade 6 from a grade 8. This guide covers the three rules you need, worked examples at exam difficulty, and the traps that cost students marks.
A surd is an irrational root — a square root (or cube root) that cannot be written as a whole number or a simple fraction. The square root of 4 is not a surd because it equals 2 exactly. The square root of 2 is a surd because its decimal expansion never terminates and never repeats.
Examiners use surds to test exact answers. If a question asks for an exact value, writing a rounded decimal loses marks. Keeping the answer in surd form is the only way to guarantee full credit.
Key distinction: √25 = 5 (not a surd, because it simplifies to a whole number). √5 remains √5 (a surd, because it cannot be simplified further).
Every simplification relies on one identity: √(ab) = √a × √b. The strategy is to factor out the largest square number from the radicand.
Step 1 — Find the largest square factor.
25 is the largest square that divides 50. Write 50 as 25 × 2. (M1)
Step 2 — Split the root.
√50 = √(25 × 2) = √25 × √2 = 5√2. (A1)
Always check: is the number under the root square-free? 2 has no square factors, so 5√2 is fully simplified.
The same logic works for multiplication. When two surds are multiplied, combine them under a single root first, then simplify.
Step 1 — Combine under one root.
√18 × √8 = √(18 × 8) = √144. (M1)
Step 2 — Evaluate the square root.
√144 = 12. (A1)
Alternative route: √18 = 3√2 and √8 = 2√2, so 3√2 × 2√2 = 6 × 2 = 12. Both methods earn full marks.
Exam boards penalise surds in the denominator. Rationalising means removing the root from the bottom of a fraction by multiplying numerator and denominator by a carefully chosen form of one.
Multiply top and bottom by the surd itself. Since √a × √a = a, the denominator becomes rational.
Use the conjugate: multiply top and bottom by (a − √b). This applies the difference of squares and eliminates the root.
Step 1 — Multiply by √5 / √5.
(3 × √5) / (√5 × √5) = 3√5 / 5. (M1)
Step 2 — Simplify.
The denominator is now rational. The answer is (3√5) / 5. (A1)
Notice the question asked for the form a√5. This means a = 3/5, which is acceptable because a is allowed to be fractional.
Step 1 — Multiply by the conjugate (2 + √3) / (2 + √3).
Numerator: 5(2 + √3) = 10 + 5√3. (M1)
Step 2 — Expand the denominator using difference of squares.
(2 − √3)(2 + √3) = 4 − 3 = 1. (M1)
Step 3 — Write the final form.
(10 + 5√3) / 1 = 10 + 5√3, so a = 10 and b = 5. (A1)
Surds can only be combined when they are like terms — meaning the number under the root is identical. You cannot add √2 and √3 directly, just as you cannot add x and y. But 4√7 + 3√7 = 7√7 works because the surd part matches.
Step 1 — Simplify each surd individually.
√27 = √(9 × 3) = 3√3.
√12 = √(4 × 3) = 2√3. (M1)
Step 2 — Collect like terms.
3√3 + 2√3 − √3 = (3 + 2 − 1)√3 = 4√3. (M1, A1)
A common error is to write √27 + √12 = √39. Roots do not add in this way. Simplify first, then combine.
Students often merge roots before simplifying. √9 + √16 equals 7, not √25. Always simplify each term separately before attempting to combine.
Even if the algebra is otherwise perfect, a surd in the final denominator typically costs the accuracy mark. Check the question: if it says “rationalise,” the denominator must be an integer.
√48 = 4√3, not 2√12. Examiners expect the largest square factor to be extracted. Factor out 16, not 4, to reach the fully simplified form in one step.
When rationalising (a − √b), the conjugate is (a + √b). Flipping the wrong sign leaves a surd in the denominator and loses method marks.
Surds are only one topic on the Higher paper. Download our free GCSE maths starter packs for AQA, Edexcel and OCR — each includes thirty worked questions across number, algebra, geometry and statistics. For structured topic-by-topic revision, see our GCSE Maths packs with board-specific topic rankings.
Revision is not only about past papers. Understanding why exact values matter — in engineering, physics and finance — can rebuild motivation. Our STEM Mission Boxes give secondary schools structured, curriculum-linked enrichment projects that use algebra, geometry and exact arithmetic in real-world contexts.