A surd is a root that cannot be written as an exact fraction — for example, √2, √3, and √50. To simplify a surd, rewrite it as the product of a square number and a remaining root: √50 = √(25 × 2) = 5√2. Always use the largest perfect-square factor to simplify in one step.
What is a surd?
A surd is an irrational number expressed as a root. √4 = 2 exactly, so it is not a surd. √5 cannot be written exactly as a decimal or fraction — it is a surd. Surds appear throughout GCSE Higher and are essential for exact answers in trigonometry and Pythagoras questions.
Key rule: √(a × b) = √a × √b. This lets you split a root into factors.
How do you simplify a surd?
Step-by-step method:
- Find the largest perfect square that is a factor of the number under the root.
- Split the surd using √(a × b) = √a × √b.
- Evaluate the square root of the perfect-square part.
- Write the simplified form.
Worked example 1 — simplify √72:
- Factors of 72 that are perfect squares: 4, 9, 36. Largest = 36.
- √72 = √(36 × 2) = √36 × √2.
- √36 = 6.
- Answer: 6√2.
Worked example 2 — simplify √200:
- Largest perfect-square factor of 200 = 100 (since 200 = 100 × 2).
- √200 = √100 × √2 = 10 × √2.
- Answer: 10√2.
| Surd | Largest sq. factor | Simplified form |
|---|---|---|
| √12 | 4 | 2√3 |
| √50 | 25 | 5√2 |
| √75 | 25 | 5√3 |
| √98 | 49 | 7√2 |
| √180 | 36 | 6√5 |
How do you add and subtract surds?
You can only add or subtract like surds — those with the same number under the root sign, just as you can only add like terms in algebra.
Worked example — simplify 3√5 + 7√5 − √5:
- All three terms have √5, so they are like surds.
- (3 + 7 − 1)√5 = 9√5.
Worked example — simplify √12 + √27:
- Simplify each surd first: √12 = 2√3; √27 = 3√3.
- 2√3 + 3√3 = 5√3.
How do you multiply surds?
Use √a × √b = √(ab), then simplify if possible.
Worked example — simplify √6 × √15:
- √6 × √15 = √(6 × 15) = √90.
- Largest perfect-square factor of 90 = 9. √90 = √(9 × 10) = 3√10.
- Answer: 3√10.
Note: (√a)² = a. For example, (√7)² = 7. This is used to rationalise denominators.
How do you rationalise the denominator?
A fraction with a surd in the denominator (e.g. 5/√3) needs to be rewritten so the denominator is rational. Multiply the top and bottom by the surd in the denominator.
Worked example — rationalise 5/√3:
- Multiply numerator and denominator by √3: (5 × √3) / (√3 × √3).
- Denominator: √3 × √3 = 3.
- Answer: 5√3 / 3.
Worked example — rationalise 6/(2√5):
- Multiply top and bottom by √5: (6√5) / (2 × 5) = 6√5 / 10.
- Simplify: 3√5 / 5.
Frequently asked questions
Why is it called a surd?
The word comes from the Latin surdus meaning "deaf" or "mute" — historically, irrational roots were considered "inaudible" to the rational number system. In modern GCSE maths, a surd simply means a root that cannot be expressed exactly as a rational number.
What if I use a smaller square factor instead of the largest?
You will still reach the correct answer, but you will need extra steps. For example, simplifying √72 as √(4 × 18) = 2√18, then noticing √18 = √(9 × 2) = 3√2, giving 2 × 3√2 = 6√2. Using the largest factor (36) does it in one step. In an exam, either route is valid.
Why must the denominator be rational in a final answer?
Convention in GCSE maths (and wider mathematics) requires "simplest form" — a surd in the denominator is considered unsimplified. More practically, it is easier to compare and calculate with fractions when the denominator is a whole number.
Are surds only square roots?
No — cube roots (∛) and higher roots can also be surds if they are irrational. However, GCSE maths focuses almost exclusively on square-root surds. At A-level and beyond you encounter cube-root surds and surd-like expressions involving nth roots.
For Socratic GCSE number and algebra practice including surds, see aitutors.me.