Surds are one of those topics that look scarier than they are. Once you know the rules, questions about surds become some of the most reliable marks you can pick up — and lose, if you haven't practised them. All three major exam boards test surds: AQA (8300), Edexcel (1MA1), and OCR (J560). On Higher tier papers, expect surds to appear on Paper 2 or 3, often within algebra or geometry questions. Foundation students should also know basic simplification.
What You Need to Know First
Before you can simplify surds confidently, make sure you're comfortable with:
- Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 — knowing these instantly is crucial
- Factors and factor pairs of a number
- Basic fraction arithmetic (you'll need it for rationalising the denominator)
- The rule √(a × b) = √a × √b
- The rule √a × √a = a
That last one catches students out constantly: √5 × √5 = 5, not 25 and not √25.
Simplifying Surds
A surd is in its simplest form when the number under the square root has no square factor other than 1. Your job is to find the largest square factor of the number, split the surd, and simplify.
The method:
- Find the largest square number that divides into the number under the root
- Rewrite the number as a product of that square number and another factor
- Apply the rule √(a × b) = √a × √b
- Simplify the square root of the square number
Worked Example 1 — Simplify √72
Step 1: Find the largest square factor of 72. Factors of 72 include: 4, 9, 36. The largest square factor is 36.
Step 2: Rewrite 72 as 36 × 2. √72 = √(36 × 2)
Step 3: Apply the rule. √(36 × 2) = √36 × √2
Step 4: Simplify. = 6√2
So √72 = 6√2. That's the simplified form.
Worked Example 2 — Simplify √200
Step 1: Largest square factor of 200. Think: 100 × 2. 100 is a square number.
Step 2: √200 = √(100 × 2)
Step 3: = √100 × √2
Step 4: = 10√2
A common wrong answer here is 5√8, which comes from choosing the factor 25 rather than 100. That's not wrong exactly — 5√8 can itself be simplified — but it means extra work and a higher chance of an error. Always look for the largest square factor first.
Adding and subtracting surds works just like collecting like terms. You can only add or subtract surds if they have the same number under the root sign.
3√5 + 7√5 = 10√5 ✓
3√5 + 7√3 = 3√5 + 7√3 (cannot simplify further) ✓
But watch out: sometimes you need to simplify first before you can collect.
Worked Example 3 — Simplify √12 + √75
Simplify each surd first: √12 = √(4 × 3) = 2√3 √75 = √(25 × 3) = 5√3
Now collect: 2√3 + 5√3 = 7√3
Rationalising the Denominator
When a fraction has a surd in the denominator, you need to rationalise it — meaning rewrite the fraction so there's no surd on the bottom. Exam mark schemes specifically expect this.
Simple case: denominator is just a surd
Multiply the top and bottom by the same surd.
Worked Example 4 — Rationalise the denominator of 6/√3
Multiply numerator and denominator by √3:
6/√3 × √3/√3 = (6√3)/(√3 × √3) = (6√3)/3 = 2√3
Harder case: denominator is of the form (a + √b) or (a − √b)
Here you use the conjugate — you change the sign in the middle.
- Conjugate of (3 + √2) is (3 − √2)
- Conjugate of (5 − √7) is (5 + √7)
Multiplying a bracket by its conjugate uses the difference of two squares: (a + b)(a − b) = a² − b².
Worked Example 5 — Rationalise 10/(5 + √3)
Conjugate of (5 + √3) is (5 − √3).
Multiply top and bottom by (5 − √3):
Numerator: 10(5 − √3) = 50 − 10√3
Denominator: (5 + √3)(5 − √3) = 25 − 3 = 22
So: (50 − 10√3)/22
Simplify by dividing everything by 2: (25 − 5√3)/11
This is a method that students find tricky at first, but once you've practised it a handful of times it becomes very mechanical.
Common Mistakes Students Make
- Not finding the largest square factor. Using √4 as a factor of √72 gives 2√18, which still needs simplifying. You'll get there, but you'll waste time and risk errors. Always hunt for the biggest square factor.
- Thinking √9 + √16 = √25. It doesn't. √9 + √16 = 3 + 4 = 7, not 5. The square root does NOT distribute over addition.
- Forgetting that √a × √a = a. Consistently writing √5 × √5 = √25 = 5 is fine — but writing it as 25 is a common slip under exam pressure.
- Not rationalising when asked. If the question says "write in the form a√b" or "give your answer in simplified surd form", leaving a surd in the denominator will cost you the accuracy mark (A mark) even if your method is correct.
- Sign errors when using conjugates. When expanding (5 + √3)(5 − √3), students sometimes forget that −√3 × √3 = −3, giving a result of 25 + 3 instead of 25 − 3. Write out every step.
Exam Tips for Surds
- The largest square factor trick is a time-saver. Write out the square numbers up to 144 at the top of your working space before starting a surds question if it helps you spot them quickly.
- Show your splitting step. Write √72 = √(36 × 2) as an explicit line. That's where the method mark (M1) usually lives — examiners want to see you've applied the rule correctly, not just the answer.
- Rationalising is always expected unless stated otherwise. If your final answer has a surd on the denominator and the question hasn't told you otherwise, rationalise it before moving on.
- AQA questions often combine surds with expanding brackets. You might see (2 + √3)² or (√5 − 1)(√5 + 2). Expand these exactly as you would any double bracket — then collect like terms and simplify.
- Check your answer makes sense. If you've simplified √48 and got 8√6, that's clearly too big (√48 ≈ 6.9). A quick sense-check against an approximation catches mistakes fast.
Practise Now
The best way to get confident with surds is to practise exam-style questions with instant feedback. Try surds questions now on GCSEMathsAI — AI-generated questions matched to your board and tier, with detailed marking and personalised feedback.