Surds explained for GCSE Maths: simplifying, multiplying and rationalising

Surds are a Higher-tier GCSE topic, and they appear most often on the non-calculator paper, where the examiner wants to see that you can manipulate exact values without reaching for a decimal. They also turn up inside other questions: an exact answer to Pythagoras, the length of a side found using trigonometry with exact values, or the solution to a quadratic left in surd form. Students who are shaky on surds therefore lose marks in places that do not obviously look like 'surd questions' at all.

This guide works through everything on the Edexcel, AQA and OCR GCSE Higher specifications for surds, in the order I teach it, with worked examples and the mistakes I see most often in mock papers.

What is a surd?

A surd is a root that cannot be simplified to a whole number or a fraction. √4 is not a surd, because it equals 2. √2, √3 and √5 are surds, because their decimal expansions go on forever without repeating — they are irrational numbers. Leaving an answer as √2 rather than 1.414… keeps it exact, which is why examiners ask for answers 'in surd form' or 'in the form a√b'.

The two rules everything is built on

Notice what is missing: there is no rule that says √a + √b = √(a + b). That is false. √9 + √16 is 3 + 4 = 7, but √25 is 5. This single misconception costs more surd marks than anything else.

Simplifying surds

To simplify a surd, find the largest square number that is a factor of the number under the root, then split the root using √(ab) = √a × √b.

1. List the square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100…
2. Find the largest one that divides exactly into the number under the root.
3. Write the surd as √(square) × √(remaining factor).
4. Square-root the square number to get the final form a√b.

For example, √72. The largest square factor of 72 is 36, so √72 = √36 × √2 = 6√2. If you use a smaller square factor first — say 4, giving 2√18 — you are not wrong, but you are not finished, because √18 still simplifies to 3√2, giving 6√2. Using the largest square factor saves time and reduces the chance of stopping too early.

A finished surd has no square factors left under the root. Always check: can the number under the root be divided by 4, 9, 16 or 25? If so, keep going.

Adding and subtracting surds

You can only add or subtract surds that have the same number under the root — think of √2 as being like x in algebra. Just as 3x + 5x = 8x, 3√2 + 5√2 = 8√2. And just as 3x + 5y cannot be simplified, 3√2 + 5√3 cannot be combined either.

The examiner's favourite twist is to give surds that look different but simplify to the same root. For example, √50 + √18. Simplify each: √50 = 5√2 and √18 = 3√2. Now they are like terms, and the answer is 8√2. If a question asks you to add surds that look unlike, simplifying first is almost always the intended route.

Multiplying surds and expanding brackets

Multiply the numbers outside the roots together and the numbers inside the roots together: 2√3 × 4√5 = 8√15. Then simplify if possible.

Expanding brackets with surds works exactly like expanding algebraic brackets — use a grid or FOIL and be systematic. Expand (3 + √2)(5 − √2). The four products are 15, −3√2, 5√2 and −√2 × √2 = −2. Collect terms: 15 − 2 + 2√2 = 13 + 2√2.

Squaring a bracket

(2 + √3)² is not 4 + 3. Write it as (2 + √3)(2 + √3) and expand fully: 4 + 2√3 + 2√3 + 3 = 7 + 4√3. Forgetting the middle terms is the same mistake students make with (x + 3)², and it is just as common with surds.

Rationalising the denominator

Rationalising means rewriting a fraction so there is no surd in the denominator. Mathematicians prefer this form because it is easier to compare and calculate with, and GCSE questions ask for it explicitly.

Case 1: a single surd in the denominator

Multiply the top and bottom by that surd. For 6/√3, multiply by √3/√3 to get 6√3/3, which simplifies to 2√3. You are multiplying by 1, so the value is unchanged — only the form changes.

Case 2: a bracket in the denominator (top of Higher tier)

If the denominator is something like 3 + √2, multiply top and bottom by the same bracket with the sign changed, 3 − √2. This uses the difference of two squares so the denominator becomes 9 − 2 = 7. For 4/(3 + √2) the result is 4(3 − √2)/7 = (12 − 4√2)/7. Expand the numerator carefully and simplify only if every term shares a common factor.

Where surds appear in other GCSE topics

• Pythagoras: a hypotenuse of √(4² + 6²) = √52 = 2√13 when an exact answer is required.
• Quadratics: solutions from the quadratic formula or completing the square, such as x = 3 ± √5.
• Exact trigonometric values: sin 60° = √3/2 and cos 45° = √2/2 (also written 1/√2).
• Area and perimeter: shapes with surd side lengths, where you must multiply or add surds.
• Proof-style questions: 'show that' a length or area equals a given surd expression.

The mistakes that cost marks

• Writing √a + √b = √(a + b). It is never true unless one of them is zero.
• Stopping simplification too early, leaving 2√18 instead of 6√2.
• Forgetting the middle terms when squaring a bracket.
• Multiplying only the denominator when rationalising — you must multiply the numerator too.
• Giving a decimal answer when the question asks for surd form or an exact value.
• Dropping a negative sign when √b × √b gives −b inside an expansion.

Surds at A-Level

Surds are assumed knowledge from the first week of A-Level Maths. They appear in quadratics, coordinate geometry, exact trigonometry, differentiation from first principles and integration answers. Students who arrive in Year 12 with surds secure have one less thing to worry about; those who do not find that small manipulation errors creep into almost every Pure topic. If you are considering A-Level Maths, getting surds fluent at GCSE is time very well spent.

A full exam-style worked example

Here is the kind of multi-step question that appears near the end of a Higher non-calculator paper. A rectangle has length (5 + √3) cm and width (5 − √3) cm. Show that its area is a whole number, and find its perimeter in the form a + b√3 where a and b are integers.

For the area, multiply the length by the width: (5 + √3)(5 − √3). Recognise the difference of two squares straight away: 25 − 3 = 22 cm². Because the surd terms cancel, the area is the whole number 22, which is exactly what the question asked you to show. Write every line, because 'show that' questions award marks for the working, not the final number.

For the perimeter, add all four sides: 2(5 + √3) + 2(5 − √3) = 10 + 2√3 + 10 − 2√3 = 20. Here the surds cancel again, so a = 20 and b = 0. Students often feel uneasy writing b = 0 and assume they have made a mistake. They have not — trust the algebra and check each step rather than changing a correct answer.

Finally, if the question went on to ask for the length of the diagonal, you would use Pythagoras: (5 + √3)² + (5 − √3)² = (28 + 10√3) + (28 − 10√3) = 56, so the diagonal is √56 = 2√14 cm. One short question has tested expanding, the difference of two squares, adding surds, Pythagoras and simplifying — which is exactly why surd fluency matters so much on the Higher paper.

How to practise surds

Surds respond well to short bursts of practice. Spend one session on simplifying alone until you can do it instantly, then one on adding and multiplying, then one on rationalising. Finish with mixed non-calculator past paper questions where surds are hidden inside Pythagoras or quadratics.

MathVault has free GCSE surd practice organised by skill, which suits the first sessions well. MathVault Premium includes full worked solutions, so you can see exactly where an expansion or rationalisation went wrong rather than only seeing the final answer.

When surds still feel confusing

If surds keep going wrong, the cause is usually underneath them: square numbers that are not automatic, or expanding brackets that is not yet reliable. In one-to-one lessons I find that underlying gap first and fix it, so the surd work then makes sense. I am a QTS fully qualified teacher and Deputy Head of Maths with over 2,000 hours of tutoring experience, and I teach GCSE and A-Level students online and in London.

The short version

1. A surd is a root that is not a whole number; it keeps answers exact.
2. √a × √b = √(ab), but √a + √b is not √(a + b).
3. Simplify by taking out the largest square factor.
4. Only like surds can be added — simplify first to make them alike.
5. Rationalise by multiplying top and bottom by the surd, or by the bracket with the sign changed.
6. Expect surds inside Pythagoras, quadratics and exact trigonometry.