GCSE Maths surds

By The STEM-Tutor team · Updated October 2026

Surds are a Higher tier topic that appears on non-calculator papers. A few rules handle almost every question.

What a surd is

A surd is a root that cannot be simplified to a whole number, such as √2 or √7. Writing answers as surds keeps them exact.

The key rules

• √a × √b = √(ab)
• √a / √b = √(a/b)
• √a × √a = a

Simplifying

Find the largest square factor: √12 = √(4 × 3) = 2√3, and √50 = 5√2.

Adding and subtracting

You can only combine like surds: 3√2 + 5√2 = 8√2. Simplify first: √8 + √18 = 2√2 + 3√2 = 5√2.

Expanding brackets

Expand as usual: (2 + √3)(1 + √3) = 2 + 2√3 + √3 + 3 = 5 + 3√3.

Rationalising the denominator

• Single surd: multiply top and bottom by the surd. 6/√3 = 6√3/3 = 2√3.
• Bracket: multiply by the conjugate. For 1/(3 + √2), multiply by (3 − √2) to get (3 − √2)/7.

Common mistakes

• Writing √a + √b = √(a + b). It is not true.
• Not simplifying the final answer fully.
• Sign errors when using the conjugate.

Why surds are useful

A surd is an exact way to write a number like √2, whose decimal goes on forever without repeating. If you round √2 to 1.41, you lose accuracy, and the error grows when you keep calculating. Keeping the surd keeps the answer exact. That is why many Higher tier questions say “give your answer in surd form” or “give your answer in the form a + b√3”. Surds also appear in Pythagoras and trigonometry, where exact side lengths often turn out to be surds.

Worked examples

Example 1: simplifying

Simplify √72.

• Find the largest square factor: 72 = 36 × 2.
• √72 = √36 × √2 = 6√2.

Example 2: adding surds

Simplify √50 + √18.

• √50 = 5√2 and √18 = 3√2.
• 5√2 + 3√2 = 8√2. You can only add like surds, just like collecting like terms in algebra.

Example 3: multiplying

Work out 2√3 × 5√6.

• Multiply the whole numbers and the surds separately: 10√18.
• √18 = 3√2, so the answer is 30√2.

Example 4: expanding brackets

Expand and simplify (3 + √5)(2 − √5).

• 6 − 3√5 + 2√5 − 5.
• = 1 − √5.

Example 5: squaring a bracket

Expand (2 + √3)².

• (2 + √3)(2 + √3) = 4 + 2√3 + 2√3 + 3.
• = 7 + 4√3. A common error is to write 4 + 3 = 7 and forget the middle terms.

Example 6: rationalising a simple denominator

Rationalise 6/√3.

• Multiply top and bottom by √3: 6√3/3.
• = 2√3.

Example 7: rationalising with a bracket

Rationalise 4/(3 − √5).

• Multiply top and bottom by the conjugate, 3 + √5.
• Denominator: 9 − 5 = 4. Numerator: 4(3 + √5).
• Answer: 3 + √5.

Example 8: surds in Pythagoras

A right-angled triangle has shorter sides 2 cm and 4 cm. Find the hypotenuse in simplest surd form.

• h² = 4 + 16 = 20, so h = √20.
• √20 = √4 × √5 = 2√5 cm.

Square numbers to know by heart

Simplifying surds is much quicker if you know the square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. When you see a surd, look for the largest of these that divides the number. If you pick a smaller square, you can still get there, but you will need to simplify again. For example, √72 = 2√18 = 2 × 3√2 = 6√2.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are surds on the Foundation paper?

Surds are mainly Higher tier. Foundation students may meet square roots, but simplifying and rationalising surds are Higher topics.

Can I use a calculator for surds?

Many calculators give surd answers, which is useful for checking. But surd questions often appear on the non-calculator paper, so learn the methods by hand.

Key takeaways

• A surd keeps an answer exact.
• √ab = √a × √b; use the largest square factor.
• Only like surds can be added or subtracted.
• Expand surd brackets carefully, including the middle terms.
• Rationalise by multiplying by the surd, or by the conjugate for brackets.

Where to go next

Surds link to Pythagoras, trigonometry and exact values.

• GCSE Maths trigonometry
• GCSE Maths topics
• GCSE Maths revision

More mistakes to avoid

• Writing √a + √b = √(a + b). For example, √9 + √16 = 7, but √25 = 5.
• Not using the largest square factor, which leaves the surd only partly simplified.
• Forgetting the middle terms when squaring a bracket.
• Multiplying only the denominator when rationalising; you must multiply the top as well.

Why rationalising works

Rationalising means removing a surd from the bottom of a fraction. It works because multiplying a fraction by something equal to 1 does not change its value. When you multiply 6/√3 by √3/√3, you are multiplying by 1, but the bottom becomes √3 × √3 = 3, a whole number. For a bracket like 3 − √5, multiplying by 3 + √5 uses the difference of two squares: (3 − √5)(3 + √5) = 9 − 5 = 4. The surd terms cancel out, which is the whole point of using the conjugate.

Where surds appear in exam questions

Surds rarely appear on their own in the hardest questions. More often they turn up inside another topic. A Pythagoras question might ask for an exact length, a trigonometry question might use exact values like √3/2, and an area question might give side lengths in surd form. Some questions combine several skills, for example finding the area of a rectangle with sides (3 + √2) and (3 − √2), which needs expanding with the difference of two squares to get 7. Being confident with the basic rules lets you focus on the bigger problem.

How to practise surds

Start with simplifying, because every other skill depends on it. Do a set of ten questions until you can spot the largest square factor quickly. Then practise adding and subtracting, then multiplying and expanding brackets, and finally rationalising. Check answers with a calculator when practising, since many calculators show surd form. Once each skill is quick, try mixed exam questions where you must choose the right method yourself.

Surds are a topic where a small amount of focused practice makes a big difference, and the marks they offer are very reliable once the rules are secure.

A quick self-test

Can you simplify √200, expand (1 + √3)² and rationalise 3/√6 without notes? If so, you are ready for exam questions. The answers are 10√2, 4 + 2√3 and √6/2.