You can only add or subtract surds that have the same surd part — just as you can only collect like algebraic terms. 3√2 + 5√2 = 8√2, but 3√2 + 5√3 cannot be simplified further. When the surd parts look different, simplify each surd first: you may discover that two surds are secretly like terms in disguise.
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.
| Expression | Like terms? |
|---|---|
| 3√5 and 7√5 | Yes — both have √5 |
| 4√3 and 4√7 | No — different surd parts |
| 2√2 and √2 | Yes — √2 = 1√2 |
| √12 and √3 | Not obviously — simplify first |
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:
- 3√7 + 2√7 = (3 + 2)√7 = 5√7
- 9√11 − 4√11 = (9 − 4)√11 = 5√11
- √5 + 6√5 = (1 + 6)√5 = 7√5
- 8√3 − 8√3 = (8 − 8)√3 = 0
When the surd parts are different, the expression is already in its simplest form:
- 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:
| Type | Example | Approach |
|---|---|---|
| Direct addition | Simplify 3√7 + 8√7 | Collect immediately |
| Simplify then add | Simplify √18 + √50 | Simplify each, then collect |
| Mixed with coefficients | Show that 5√12 − √27 = 7√3 | Simplify, collect, verify equals stated answer |
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.
For Socratic GCSE surd practice with Professor Pi, see aitutors.me.