A surd is a root of a rational number that is itself irrational, such as √2, √5 or √72. Surds let you write answers exactly where a decimal can only approximate: √2 = 1.41421356… never terminates or repeats, so writing 1.41 already introduces a rounding error.
This is why A-Math questions so often say “without using a calculator” or “leave your answer in the form a + b√c”. The examiner is testing whether you can manipulate exact values, and a decimal answer scores zero even if it is numerically close.
What these notes cover
- •What surds are and why they matter
- •The two laws of surds
- •Simplifying surds
- •Adding and subtracting like surds
- •Multiplying and dividing surds
- •Expanding and squaring brackets
- •Conjugates and the difference of two squares
- •Rationalising the denominator
- •Equating rational and irrational parts
- •Surds in geometry: exact lengths, areas and volumes
- •Common mistakes and exam technique
- •Summary and pre-exam checklist
Key definitions
- Surd
- A root of a rational number that is itself irrational. It cannot be written as a fraction of two integers, and its decimal expansion neither terminates nor recurs. √2 is a surd; √9 = 3 is not.
- Simplest form
- A surd is in simplest form when the number under the root has no perfect square factor other than 1, e.g. 6√2 rather than 2√18.
- Like surds
- Surds that have the same number under the root sign after simplification, e.g. 3√2 and 5√2. Only like surds can be added or subtracted.
- Conjugate
- The expression obtained from a two-term expression by flipping the sign between the two terms; nothing else changes. The conjugate of a + √b is a − √b, and the conjugate of 5 − 2√3 is 5 + 2√3.
- Rationalising the denominator
- Rewriting a fraction so that its denominator becomes a rational number, with no surd underneath, without changing the value of the fraction.
Key formulas
Multiplication law
For a ≥ 0 and b ≥ 0. There is no addition law: √(a + b) ≠ √a + √b.
Simplifying a surd
For a > 0. Extract the largest perfect square factor, e.g. √72 = √(36 × 2) = 6√2.
Perfect square with a surd
Write out all three terms. The middle term 2a√b is where the surd survives; if your squared answer has no surd, the middle term was dropped.
Rationalising a single surd
Multiply top and bottom by √b. This multiplies the fraction by 1, so its value is unchanged, e.g. 6 ÷ √3 = 6√3 ÷ 3 = 2√3.
Conjugate product (difference of two squares)
The product is rational, which is how a two-term denominator is rationalised. The coefficient b is squared too: (3 − 2√5)(3 + 2√5) = 9 − 4 × 5 = −11.
Equating rational and irrational parts
Valid only when a, b, c and d are rational. The same principle works for √3, √5 or any other surd.
Worked example
Find the rational numbers a and b such that (a + b√3)(2 − √3) = 7 − 3√3.
- 1Expand the left-hand side fully: (a + b√3)(2 − √3) = 2a − a√3 + 2b√3 − b√3 × √3.
- 2Since √3 × √3 = 3, the last term is −3b, so the left-hand side is 2a − 3b − a√3 + 2b√3.
- 3Collect the rational and √3 terms: (2a − 3b) + (−a + 2b)√3.
- 4Since a and b are rational, equate the rational parts: 2a − 3b = 7 …(1).
- 5Equate the coefficients of √3: −a + 2b = −3 …(2).
- 6Multiply (2) by 2 to get −2a + 4b = −6 …(3), then add (1) and (3) to eliminate a, giving b = 1.
- 7Substitute b = 1 into (1): 2a − 3 = 7, so 2a = 10 and a = 5.
- 8Check: (5 + √3)(2 − √3) = 10 − 5√3 + 2√3 − 3 = 7 − 3√3, which matches the right-hand side, so a = 5 and b = 1.
Watch out: Never divide both sides by √3 or merge the two parts into a single equation. One surd equation gives exactly two ordinary equations, never one.
Common mistakes
Splitting a root over addition
Wrong: √(a + b) = √a + √b, so √(9 + 16) = 3 + 4 = 7
Right: √(9 + 16) = √25 = 5. A root splits over multiplication and division only, never over + or −.
Dropping the middle term when squaring
Wrong: (3 + √2)² = 9 + 2 = 11
Right: (3 + √2)² = 3² + 2(3)(√2) + (√2)² = 9 + 6√2 + 2 = 11 + 6√2
Stopping before the simplest form
Wrong: √72 = √(4 × 18) = 2√18
Right: √72 = √(36 × 2) = 6√2. Extract the largest perfect square factor, or keep simplifying until no square factor remains.
Questions students ask
What is a surd in A-Math?
A surd is a root of a rational number that is irrational, such as √2 or √5; it cannot be written as a fraction of two integers. Roots that simplify to rational numbers, like √9 = 3, are not surds.
How do you simplify a surd like √72?
Find the largest perfect square factor of the number under the root, split it off using √(ab) = √a × √b, and take its square root outside: √72 = √(36 × 2) = 6√2. If you pick a smaller square factor such as 4, keep simplifying until no square factor remains.
How do you rationalise a denominator with two terms?
Multiply the top and bottom by the conjugate, which flips the sign between the two terms. The denominator becomes a difference of two squares, which is rational. Keep the terms in their original order, so (2√3 − 5)(2√3 + 5) = 12 − 25 = −13.
Why can you equate rational and irrational parts in a surd equation?
If a + b√2 = c + d√2 with b ≠ d, then √2 = (c − a) ÷ (b − d), which would be rational — a contradiction, so b = d and then a = c. This only works when the question states that the unknowns are rational or integers.
In the full PDF
- •Labelled diagrams: rectangle ABCD with a surd length and a right-angled triangle with surd legs
- •Further worked examples on simplifying, expanding brackets and rationalising, including a denominator that comes out negative
- •Exact geometry examples: missing breadth of a rectangle, hypotenuse with surd legs and the volume of a cuboid
- •The remaining common mistakes, including forgetting to rationalise and sign slips with the conjugate
- •Exam technique for aiming at a required form and checking by multiplying back
- •Five-skill summary and final pre-exam checklist