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This is a Higher-tier [H] lesson throughout — every section below is Higher-only content in OCR GCSE Mathematics (J560). A surd is a root that cannot be simplified to a rational number, such as 2 or 12. Surds let us give exact answers rather than rounded decimals, which is why "leave your answer in surd form" is a common instruction on the Higher papers. This lesson covers simplifying surds, the rules for surd arithmetic (multiplying, dividing, adding and subtracting), and rationalising a denominator.
This lesson mainly builds AO1 fluency in manipulating surds, with AO2 reasoning when you justify that a number is irrational, and AO3 problem-solving where surds arise from areas, Pythagoras or expanding brackets.
| Term | Meaning |
|---|---|
| Surd | A root that is irrational, e.g. 2, 12, 35 |
| Rational number | A number expressible as a fraction of integers |
| Irrational number | A number that cannot be written as such a fraction |
| Simplify a surd | Write it with the smallest possible number under the root |
| Rationalise | Remove a surd from the denominator of a fraction |
| Like surds | Surds with the same number under the root, e.g. 32 and 52 |
2=1.41421356… never terminates and never recurs, so any decimal we write down is only an approximation. Leaving the answer as 2 keeps it exact. The two rules that drive all surd work are:
a×b=ab,ba=ba.
Note that there is no such rule for addition: a+b=a+b in general.
A quick way to see which roots are surds: a square root is a surd exactly when the number under it is not a perfect square. So 9=3 is not a surd (it is a whole number), but 10 is a surd because 10 lies between the perfect squares 9 and 16 and has no exact square root. The same idea works for cube roots and perfect cubes. Recognising this stops you wasting time trying to "simplify" a root that is already rational, and tells you immediately when an answer should be left in surd form.
To check the addition rule fails, test it with numbers: 9+16=3+4=7, but 9+16=25=5. Since 7=5, there is clearly no "add under the root" rule — a useful sanity check whenever you are tempted to combine surds incorrectly.
To simplify a surd, find the largest square factor of the number under the root, then split it off.
Simplify 12.
The largest square factor of 12 is 4 (12=4×3).
12=4×3=4×3=23.
Answer: 23.
Simplify 72.
The largest square factor of 72 is 36 (72=36×2).
72=36×2=62.
Answer: 62.
Common error: Using a smaller square factor, e.g. 72=4×18=218. This is true but not fully simplified — 18 simplifies again. Always take the largest square factor.
Write 53 as a single square root.
This is the reverse of simplifying: take the number outside back inside by squaring it. 5=25, so 53=25×3=25×3=75.
Answer: 75. This reverse skill is handy when you need to compare two surds — for example, is 53 bigger than 45? Writing both as single roots, 75 and 80, shows at once that 45=80 is the larger.
Use a×b=ab and simplify the result; multiply any whole-number parts separately.
Work out 6×8, giving your answer in simplest surd form.
6×8=48=16×3=43.
Answer: 43.
Work out 35×210.
Multiply whole numbers and surds separately: (3×2)(5×10)=650=6×52=302.
Answer: 302.
Work out 540.
540=540=8=22.
Answer: 22.
You can only add or subtract like surds (same number under the root) — treat the surd like an algebraic term. Simplify each surd first; that often turns unlike-looking surds into like ones.
Simplify 50+18.
Simplify each: 50=52 and 18=32. Now they are like surds:
52+32=82.
Answer: 82.
Common error: Writing 50+18=68. There is no "add under the root" rule — you must simplify to like surds and add the coefficients.
It is conventional to write fractions with no surd in the denominator. For a single surd, multiply top and bottom by that surd.
Rationalise 36.
Multiply numerator and denominator by 3:
36×33=363=23.
Answer: 23.
Why does this work? Multiplying the denominator 3 by itself gives 3×3=9=3, a rational number, which clears the surd from the bottom. We multiply the top by the same 3 so that we are really multiplying the whole fraction by 33=1, leaving its value unchanged.
Rationalise 85, giving your answer in simplest form.
Multiply top and bottom by 8: 85×8=840. Now simplify the surd: 40=210, so the fraction is 8210=410.
Answer: 410.
Common error: Forgetting to simplify 40 and to cancel the 82 down to 41. "Simplest form" requires both steps.
Expand and simplify 3(6+23).
Multiply through: 3×6=18=32, and 3×23=2×3=6.
3(6+23)=32+6.
Answer: 32+6.
Expand and simplify (5+2)(5−2).
This is (a+b)(a−b)=a2−b2 with a=5, b=2:
(5)2−22=5−4=1.
Answer: 1 (a rational number — the surds cancel).
This is the idea behind rationalising a denominator that is a sum containing a surd: multiply by the conjugate.
Rationalise 3+54.
Multiply top and bottom by the conjugate 3−5:
3+54×3−53−5=(3)2−(5)24(3−5)=9−512−45=412−45=3−5.
Answer: 3−5.
A right-angled triangle has legs of length 2 cm and 5 cm. Work out the exact length of the hypotenuse.
By Pythagoras, h2=22+(5)2=4+5=9, so h=9=3 cm.
Answer: 3 cm (here the surd disappears).
Simplify 218+50, giving your answer as an integer.
Simplify the top first: 18=32 and 50=52, so the numerator is 32+52=82.
282=8.
Answer: 8. The 2 in the numerator and denominator cancel, leaving a whole number — a neat reminder that surds often simplify away entirely.
A square has area 48 cm². Work out the exact length of one side, in simplest surd form.
The side is 48. Simplify: 48=16×3=43.
Answer: 43 cm. Leaving it as 43 is exact; writing 6.93 cm would only be an approximation and could lose an "exact value" mark.
Specimen question modelled on the OCR J560 paper format: Simplify fully 75−27.
Grades 3–4 response: 75=53 and 27=33, so 53−33=23.
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