Mathematics 4024 · Syllabus tracker
Surds
Simplifying surds, adding, multiplying and expanding them, and rationalising denominators.
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Quick revision
Surds on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Exact roots: simplifying, calculating with surds, and rationalising denominators.
1Simplifying surds
- A surd is a root that is irrational, like √2 or √12. It is exact — keep it as a surd unless told to round.
- Find the largest square factor: √72 = √(36 × 2) = 6√2.
- Square factors to look for: 4, 9, 16, 25, 36, 49, 64, 81, 100.
Formulas
Worked example
Simplify √75 + √12.
- √75 = √25 × √3 = 5√3.
- √12 = √4 × √3 = 2√3.
- 5√3 + 2√3.
7√3
2Adding, multiplying and expanding
- Only like surds add or subtract: 6√2 − 4√2 = 2√2, but √2 + √3 cannot be combined.
- Multiply: √3 × √12 = √36 = 6; 2√5 × 3√2 = 6√10.
- Expand brackets just as in algebra, then simplify each term.
- (a + √b)(a − √b) = a2 − b: the surds cancel. This is used to rationalise.
Worked example
Expand and simplify (3 + √2)2.
- (3 + √2)(3 + √2) = 9 + 3√2 + 3√2 + 2.
- Collect: 11 + 6√2.
11 + 6√2
Exam tip: (a + √b)2 is NOT a2 + b — there is a middle term 2a√b.
3Rationalising the denominator
- 'Rationalise' means: no surd on the bottom of the fraction.
- a/√b: multiply top and bottom by √b. 10/√5 = 10√5/5 = 2√5.
- a/(b + √c): multiply top and bottom by (b − √c). a/(b − √c): multiply by (b + √c).
How to do it
- Look at the bottom.
- Single surd √b → multiply top and bottom by √b.
- b ± √c → multiply top and bottom by the same with the opposite sign.
- Expand and simplify; cancel any common factor.
Worked example
Rationalise 6/(3 − √3).
- Multiply by (3 + √3)/(3 + √3).
- Bottom: 9 − 3 = 6.
- Top: 6(3 + √3).
- 6(3 + √3)/6.
3 + √3
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| √9 + √16 = √25 | √9 + √16 = 3 + 4 = 7 |
| Leaving √72 as 2√18 | Use the largest square factor: 6√2 |
| (3 + √2)2 = 9 + 2 | Middle term: 9 + 6√2 + 2 = 11 + 6√2 |
| Multiplying only the top when rationalising | Multiply top and bottom by the same surd |
★In the exam
- 'Give your answer in the form a + b√c' — make sure a, b, c are integers and the surd is simplified.
- Surd questions are on the non-calculator paper; check by squaring your answer mentally.
- Exact answers (like in Pythagoras) are often wanted as simplified surds.