Adding and Subtracting Surds: GCSE Maths Step-by-Step

What are like surd terms?

A surd has two parts: a coefficient (the number in front) and a surd part (the square root). Like surd terms have the same surd part.

You add or subtract like surds by adding or subtracting their coefficients, leaving the surd part unchanged. This is identical to collecting like terms in algebra (3x + 5x = 8x).

How do you add and subtract like surds directly?

Worked examples:

1. 3√7 + 2√7 = (3 + 2)√7 = 5√7
2. 9√11 − 4√11 = (9 − 4)√11 = 5√11
3. √5 + 6√5 = (1 + 6)√5 = 7√5
4. 8√3 − 8√3 = (8 − 8)√3 = 0

When the surd parts are different, the expression is already in its simplest form:

5. 4√2 + 3√5 → cannot be simplified (different surd parts)

How do you simplify surds before adding or subtracting?

The key step is to express each surd in its simplest form: find the largest perfect square factor and pull it out.

Reminder: √(a × b) = √a × √b. If a is a perfect square, √a is a whole number.

Worked example 1: Simplify √8 + √2.

√8 = √(4 × 2) = √4 × √2 = 2√2

So: √8 + √2 = 2√2 + √2 = (2 + 1)√2 = 3√2

Worked example 2: Simplify √12 + √75 − √27.

• √12 = √(4 × 3) = 2√3
• √75 = √(25 × 3) = 5√3
• √27 = √(9 × 3) = 3√3

Expression: 2√3 + 5√3 − 3√3 = (2 + 5 − 3)√3 = 4√3

Worked example 3: Simplify 3√50 − 4√32 + √2.

• √50 = √(25 × 2) = 5√2, so 3√50 = 15√2
• √32 = √(16 × 2) = 4√2, so 4√32 = 16√2
• √2 = √2

Expression: 15√2 − 16√2 + √2 = (15 − 16 + 1)√2 = 0

That result — zero — is exact, not a rounding. Checking: 3√50 ≈ 21.21, 4√32 ≈ 22.63, √2 ≈ 1.41; 21.21 − 22.63 + 1.41 ≈ 0. ✓

What if there are coefficients alongside the surds?

Treat coefficients and surd parts separately. Multiply the integer coefficient into the coefficient of the surd first.

Worked example: Simplify 2√18 + 3√8.

• 2√18 = 2 × √(9 × 2) = 2 × 3√2 = 6√2
• 3√8 = 3 × √(4 × 2) = 3 × 2√2 = 6√2

Expression: 6√2 + 6√2 = 12√2

Worked example: Simplify 5√45 − 2√20 + √5.

• 5√45 = 5 × 3√5 = 15√5
• 2√20 = 2 × 2√5 = 4√5
• √5 = 1√5

Expression: 15√5 − 4√5 + √5 = (15 − 4 + 1)√5 = 12√5

What does an exam question on this topic look like?

Exam questions on surd addition/subtraction are usually one of three types:

For "show that" questions, work from the left-hand side step by step until you reach the right-hand side. Do not work backwards from the answer.

Frequently asked questions

Can I add surds with different square roots?

Only if they simplify to the same surd part. √2 and √3, in their simplest forms, have different surd parts and cannot be combined. But √8 and √2 look different yet √8 simplifies to 2√2, so they can be combined.

Do I ever need to add surds involving cube roots or higher?

At GCSE, surd questions involve square roots only. Cube roots appear only in the context of evaluating expressions like ∛8 = 2 (a rational result). Rules for adding cube-root surds follow the same like-terms logic, but that is beyond GCSE scope.

How do I check my simplified surd answer?

Convert each original surd to a decimal using a calculator, compute the decimal result, then convert your simplified surd to a decimal and check they match. For example: √8 + √2 ≈ 2.828 + 1.414 = 4.243; 3√2 ≈ 3 × 1.414 = 4.243. ✓

Why can't you just add the numbers inside the square roots?

√8 + √2 ≠ √(8 + 2) = √10. This is a very common error. The square root does not distribute over addition. √10 ≈ 3.162, but the correct answer 3√2 ≈ 4.243 — very different. Always simplify surds individually before combining them.

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For Socratic GCSE surd practice with Professor Pi, see aitutors.me.