Surds and rationalising explained: GCSE maths higher

Surds come up on every higher tier GCSE Maths paper, and they are worth easy marks once you know the rules. The topic boils down to a handful of techniques: Recognising a surd, simplifying it, combining surds by adding or subtracting, multiplying them, and rationalising denominators.

This guide walks through each technique with worked examples so you can see exactly what to do – and, just as importantly, why each step works.

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. √9 is not a surd because it equals 3. But √2, √3, √5 and √7 are all surds – their decimal expansions go on forever without repeating.

Surds are irrational numbers. In GCSE Maths you almost always work with square roots, although cube roots can appear occasionally. The reason examiners like surds is that they give exact answers. Writing √2 is precise; writing 1.41421... is not.

To simplify a surd, you look for a square number that is a factor of the number under the root sign. The key rule is √(a × b) = √a × √b.

Take √12 as an example. 12 = 4 × 3, and 4 is a square number. So √12 = √4 × √3 = 2√3.

Another example: √72. The largest square factor of 72 is 36, because 72 = 36 × 2. So √72 = √36 × √2 = 6√2.

Always look for the largest square factor if you can spot it. If you use a smaller one, you will need to simplify again. For instance, √72 = √4 × √18 = 2√18, but √18 simplifies further to 3√2, giving 2 × 3√2 = 6√2. You reach the same answer – it just takes an extra step.

You can only add or subtract surds that have the same number under the root sign – just like collecting like terms in algebra.

3√5 + 7√5 = 10√5. Think of it as 3 lots of √5 plus 7 lots of √5.

6√2 – 2√2 = 4√2.

If the surds look different, simplify them first and then check again. For example, √12 + √27 looks impossible at first. But √12 = 2√3 and √27 = 3√3, so √12 + √27 = 2√3 + 3√3 = 5√3.

You cannot add surds with different roots. 2√3 + 4√5 stays as 2√3 + 4√5 – there is no simpler form.

The core rule for multiplying surds is √a × √b = √(a × b). So √3 × √5 = √15.

When surds have coefficients, multiply the coefficients together and the surds together. For example, 2√3 × 4√5 = (2 × 4) × (√3 × √5) = 8√15.

A special case to remember: √a × √a = a. This follows directly from the definition of a square root. So √7 × √7 = 7. This fact is central to rationalising denominators, which is the next section.

Expanding brackets works the same way as with algebra. Multiply each term in one bracket by each term in the other.

(2 + √3)(4 – √3) = 2 × 4 + 2 × (–√3) + √3 × 4 + √3 × (–√3) = 8 – 2√3 + 4√3 – 3 = 5 + 2√3.

Notice how √3 × √3 became 3. That simplification always happens when a surd is multiplied by itself, and it is the whole reason rationalising works.

Rationalising the denominator means rewriting a fraction so there is no surd on the bottom. The value of the fraction does not change – you are simply writing it in a form that examiners (and mathematicians) consider tidier.

There are two cases you need to know for GCSE.

When the denominator is a single surd like √a, multiply the top and bottom of the fraction by √a.

Example: Rationalise 5 / √3.

Multiply numerator and denominator by √3. That gives (5 × √3) / (√3 × √3) = 5√3 / 3. Done.

Another example: 4 / 3√2. Multiply top and bottom by √2: (4√2) / (3 × 2) = 4√2 / 6 = 2√2 / 3.

When the denominator has two terms, such as 3 + √5 or √2 – 1, you multiply top and bottom by the conjugate. The conjugate is the same expression with the sign between the two terms flipped.

The conjugate of (3 + √5) is (3 – √5). The conjugate of (√2 – 1) is (√2 + 1).

This works because of the difference of two squares: (a + b)(a – b) = a² – b². When one of the terms is a surd, squaring it removes the root.

Example: Rationalise 6 / (3 + √5).

Multiply top and bottom by (3 – √5). The denominator becomes (3 + √5)(3 – √5) = 9 – 5 = 4. The numerator becomes 6(3 – √5) = 18 – 6√5. So the answer is (18 – 6√5) / 4, which simplifies to (9 – 3√5) / 2.

Question: Simplify (3 + √2)(3 – √2), then use your answer to rationalise 10 / (3 + √2). Give your answer in the form a + b√2.

Step 1: Expand (3 + √2)(3 – √2). Using the difference of two squares, this is 3² – (√2)² = 9 – 2 = 7.

Step 2: Rationalise 10 / (3 + √2). Multiply top and bottom by the conjugate (3 – √2). The denominator is 7 (from step 1). The numerator is 10(3 – √2) = 30 – 10√2.

Step 3: Write in the required form. The fraction is (30 – 10√2) / 7, which is 30/7 – 10√2/7. In the form a + b√2, a = 30/7 and b = –10/7.