For students and parents · O-Level A-Math
Surds, Rationalising and the Exact-Form Discipline
Roots that do not simplify, kept in exact form.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
A surd is a root that does not come out evenly, like √(2) or √(3). A-Math wants these left in exact form, not turned into decimals, because rounding loses accuracy and, in this part of the syllabus, the mark. The two skills tested are simplifying a surd and clearing a surd out of a denominator, and both rest on one idea: a surd times itself gives a whole number.
Simplifying a surd
Pull out the largest perfect-square factor. √(50) = √(25 x 2) = √(25) x √(2) = 5 √(2). The skill is spotting that 50 contains the square factor 25; a surd is fully simplified when nothing left under the root has a square factor.
Rationalising a single surd
A denominator should not be left as a surd. Multiply top and bottom by that surd, which turns the bottom into a whole number:
1/√(2) = (1 x √(2))/(√(2) x √(2)) = √(2)/2.
Another: 6/√(3) = (6 x √(3))/(√(3) x √(3)) = 6 √(3)/3 = 2 √(3). The denominator is now whole, and the answer is exact.
Rationalising with a conjugate
When the denominator is a sum or difference like 2 + √(3), multiply by its conjugate 2 - √(3). This uses the difference of two squares to remove the surd:
(2 + √(3))(2 - √(3)) = 4 - 3 = 1.
So 1/(2 + √(3)) = (2 - √(3))/((2 + √(3))(2 - √(3))) = (2 - √(3))/1 = 2 - √(3). The conjugate is the same two terms with the middle sign flipped, and it is what makes the cross terms cancel. As a clean illustration of the pattern, (3 + √(5))(3 - √(5)) = 9 - 5 = 4.
The teaching point
Every surd skill rests on one fact, that a surd times itself, or a sum times its conjugate, gives a whole number, and on one discipline, keeping the answer exact to the end. The marks here are lost by decimalising early or by leaving a surd in the denominator. Simplify by pulling out square factors, rationalise single surds by multiplying by the surd, and rationalise sums by multiplying by the conjugate.
If your child reaches for the calculator when a question wants an exact surd answer, that is a habit to retrain, and it sits under a lot of later A-Math. GPA's Secondary Math programme builds the exact-form discipline on real papers. A short diagnostic consult will show where the rounding creeps in.
Every surd manipulation on this page was checked independently before publishing.