NumberHigher only★ Higher signature
Surds.
Simplifying, operating, rationalising.
A surd is a root that can't be simplified to a rational — like √2, √3, √5. Keep them exact by never rounding. Three skills: simplify, operate, rationalise.
I · Key definitions
Surd
A root that can't be simplified to a rational number.
√2, √3, √15
Rational
A number that can be written as a fraction.
√16 = 4 is NOT a surd
Simplified
No perfect square factor left under the root.
√72 = 6√2
Rationalised
No surd in denominator of a fraction.
3/√5 = 3√5/5
Conjugate
Flip the sign between two terms.
conjugate of (2+√3) is (2−√3)
Like surds
Same number under the root — can be added/subtracted.
5√2 + 3√2 = 8√2
II · The six surd rules
√a × √b = √(ab)
multiply under one root
√a ÷ √b = √(a/b)
divide under one root
a√c + b√c = (a+b)√c
like surds add/subtract
(√a)² = a
square removes the root
√(a²b) = a√b
pull out perfect square
√a · √a = a
same surd × self = the number
III · Simplifying — the method
Step-by-step
1. Find biggest square factor of number
2. Split: √72 = √(36 × 2)
3. √36 comes out: 6√2
√72 → 36×2 → 6√2
Useful perfect squares
4, 9, 16, 25, 36, 49, 64, 81, 100
check divisibility by biggest first
Rationalising (simple)
Multiply top + bottom by √b
3/√5 = 3√5/(√5·√5) = 3√5/5
Rationalising (conjugate)
For 1/(a + √b):
× by conjugate (a − √b)
top + bottom
2/(1+√3) × (1−√3)/(1−√3) = (2(1−√3))/(1−3) = √3 − 1
IV · Worked examples
Simplify √200 as fully as possible.
i.Largest square factor of 200 = 100
ii.√200 = √(100 × 2) = √100 × √2
iii.= 10√2
√200 = 10√2
Add and subtract like surds
3 marks
Simplify √50 + √18 − √8.
i.Simplify each first: √50 = 5√2, √18 = 3√2, √8 = 2√2
ii.Now all are like surds: 5√2 + 3√2 − 2√2
iii.Combine coefficients: (5 + 3 − 2)√2 = 6√2
√50 + √18 − √8 = 6√2
Rationalise with a conjugate
3 marks
Rationalise the denominator of 6 / (3 − √5).
i.Multiply top + bottom by conjugate (3 + √5)
ii.Top: 6 × (3 + √5) = 18 + 6√5
iii.Bottom: (3 − √5)(3 + √5) = 9 − 5 = 4 (diff of squares)
iv.= (18 + 6√5) / 4 = (9 + 3√5) / 2
6/(3−√5) = (9 + 3√5)/2
V · Common mistakes & examiner tips
Common mistakes
Saying √a + √b = √(a+b). NO. Only works for multiplication.
Leaving √a in the denominator. Must rationalise for full marks.
Not simplifying fully. √50 = 5√2, not just √50.
Expanding (a+√b)(a−√b) wrongly. It's a² − b (diff of squares).
Rounding early — surds keep answers exact; don't convert to decimals.
Examiner tips
Always check for perfect square factors — √12, √18, √50 all simplify.
Memorise perfect squares up to 225 — speeds simplification.
For rationalising, always use conjugate if denominator has two terms.
Show your method — 'multiply by √5/√5' explicitly earns method marks.
Check your answer by squaring — should give a rational number.