Sheet № 79 · Higher only · AQA · Edexcel · OCR
Simplifying Surds –
Simplifying surds is a key Higher tier topic that appears regularly on AQA, Edexcel, and OCR GCSE Maths papers. A surd is an irrational root that cannot be simplified to a whole number, and exam questions expect you to leave answers in exact surd form rather than as decimals.
§Key definitions
Simplifying a surd
means rewriting it so the number under the root sign is as small as possible. You do this by finding the largest square number that is a factor of the number under the root. For example, √48 = √(16 × 3) = √16 × √3 = 4√3.
Question:
Simplify √72.
Answer:
Shown: (2√3)² + (√5)² = 12 + 5 = 17
Q1 (Higher):
Simplify √200.
Q2 (Higher):
Simplify 3√12 − √27.
§Formulas to memorise
√(ab) = √a × √b — you can split a root into the product of two roots
√a × √a = a — a surd multiplied by itself gives the number under the root
p√n + q√n = (p + q)√n — add surds with the same root like collecting like terms
Split the surd: √(largest square × remaining) = √(square) × √(remaining).
To multiply surds, use √a × √b = √(ab), then simplify the result.
Worked example
Simplify √72.
Working:
⚠ Common mistakes
- ✗Not finding the largest square factor. Writing √72 = √(4 × 18) = 2√18 is not fully simplified. Always look for the largest square factor, or continue simplifying until the number under the root has no square factors left.
- ✗Adding surds with different roots. You cannot add √2 + √3 — they are unlike terms. You can only combine surds when the number under the root is the same.
- ✗Forgetting to square the coefficient. When squaring 3√5, both parts must be squared: (3√5)² = 9 × 5 = 45, not 3 × 5 = 15.
✦ Exam tips
- →Memorise the square numbers up to 225 (15²) so you can spot square factors quickly in the exam.
- →When simplifying a surd, if you do not immediately spot the largest square factor, you can simplify in stages: √72 = √(4 × 18) = 2√18 = 2√(9 × 2) = 2 × 3√2 = 6√2.
- →"Give your answer in surd form" means you must not use a calculator to write a decimal. Leave roots in place.