Surds
Simplifying exact roots and rationalising denominators.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 1.6.1Simplify surds and rationalise denominators.Supplement
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Surds
A surd is a root that cannot be written exactly as a fraction — √2, √3, √5. Leaving an answer in surd form keeps it exact, whereas a decimal is only an approximation.
The rules
√a × √b = √(ab)√a / √b = √(a/b)√a × √a = a
Note there is no rule for √(a + b) — it does not equal √a + √b. Testing with 9 and 16 makes this obvious: √25 = 5, but 3 + 4 = 7.
Simplifying
Look for a factor that is a perfect square: √50 = √(25 × 2) = √25 × √2 = 5√2 Always take out the largest square factor, or you will have to simplify twice.
Adding and subtracting
Only like surds combine, exactly like algebraic terms: 3√2 + 5√2 = 8√2, but 3√2 + 5√3 cannot be simplified. Sometimes simplifying first reveals like terms: √8 + √2 = 2√2 + √2 = 3√2.
Rationalising the denominator
Convention says a surd should not be left on the bottom of a fraction. Multiply top and bottom by the surd: 3/√5 = (3 × √5)/(√5 × √5) = 3√5/5 For a denominator such as 2 + √3, multiply by its conjugate 2 − √3, because (2 + √3)(2 − √3) = 4 − 3 = 1 — the surds cancel through the difference of two squares.
Think of it like this
A surd is an exact fingerprint of a number; the decimal is a blurred photocopy. Rationalising is tidying rather than changing — the value is identical, only the presentation improves.
Worked examples
Method, step by step
Simplify √75 + √12.
- 1√75 = √(25 × 3) = 5√3.
- 2√12 = √(4 × 3) = 2√3.
- 3These are like surds, so add the coefficients: 5√3 + 2√3 = 7√3.
7√3
Rationalise the denominator of 6/√3.
- 1Multiply top and bottom by √3: (6 × √3)/(√3 × √3).
- 2The denominator becomes 3.
- 3= 6√3/3 = 2√3.
2√3
Common misconceptions
- Writing `√(a + b) = √a + √b`. This is false; only multiplication and division split.
- Adding unlike surds, e.g. claiming `√2 + √3 = √5`.
- Failing to take out the largest square factor, e.g. writing `√50 = 2√12.5` instead of `5√2`.
In the exam
- If a question says "give your answer in exact form" or "in surd form", do not use a calculator to decimalise it.
- For a two-term denominator, multiply by the conjugate — same terms, opposite sign — and the surds vanish.