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Cambridge IGCSE (CIE) · International Maths Extended
Surds
Surds are square roots that can't be turned into a "nice" number — and this chapter is really just about learning the legal moves you're allowed to make with them so you can simplify them, combine them, and get rid of them from the bottom of a fraction, all without ever losing exact accuracy.
Summary — The Whole Chapter at a Glance
A surd is the square root of a positive integer that isn't a perfect square (e.g. √2, √13 — but NOT √16, since that's just 4).
Surds let us write exact values instead of rounded decimals — this matters a lot for accuracy in exams.
You can multiply and divide numbers under square roots together: √a × √b = √(ab), and √a ÷ √b = √(a÷b).
You cannot add or subtract numbers under square roots — √9 + √4 ≠ √13. This is the single most common mistake in this topic.
Simplifying a surd means pulling out the largest perfect-square factor hiding inside it, e.g. √48 = 4√3.
You can add/subtract surds only once they're "like terms" — same number under the root — just like collecting algebraic terms.
Rationalising the denominator means rewriting a fraction so there's no surd left on the bottom.
For a single surd on the bottom, multiply top and bottom by that surd. For an expression like (a + √b), multiply by its "conjugate" (a − √b), using the difference of two squares.
1 · What Is a Surd, and Why Bother?
Picture your calculator. If you type in √16, it happily gives you 4 — clean, exact, done. But if you type in √2, it spits out something like 1.414213562... and keeps going forever. That's because √2 is irrational — it can never be written as a neat fraction or a terminating/repeating decimal. No matter how many digits your calculator shows you, it's always an approximation, never the exact value.
This is exactly the problem surds solve. Instead of writing the rounded, slightly-wrong decimal, we just leave the answer as √2. It looks less "finished" than a decimal, but it's actually more accurate — it's the exact value, with zero rounding error.
A surd = the square root of a positive integer that is NOT a perfect square
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... — their square roots are whole numbers, so they are NOT surds.
Spotting surds vs non-surds
This trips people up more than you'd expect, because it's not really about how "ugly" the number looks — it's about a precise definition.
These ARE surds
Why
√2, √13, √99, √201
None of 2, 13, 99, or 201 are perfect squares, so their square roots can't simplify to a whole number.
These are NOT surds
Why
√16
16 is a perfect square (4²), so √16 = 4 exactly — a whole number, not a surd.
√(2/3)
2/3 isn't an integer at all — surds are specifically defined using integers.
√6.23
6.23 isn't an integer either — same reason.
Think of it like this
A surd is basically a square root that "doesn't clean up." If a perfect square is hiding inside the number (even partially), we can rescue part of it — but if there's genuinely no perfect square factor at all, the whole thing stays trapped under the root sign forever, in its most simplified form.
Practice Question
Which of the following are surds: √25, √50, √(9/4), √0.5, √11?
▶ Show Answer
2 · The Four Legal Moves With Surds
Before we simplify anything, you need to know exactly what you're allowed to do to surds — because the rules are stricter than they look, and the exam loves testing the one thing you're not allowed to do.
Move 1 — Multiplying surds
√a × √b = √(a × b)
You can combine numbers under square roots when multiplying.
Worked Example
√3 × √5 = √(3×5) = √15
Move 2 — Dividing surds
√a ÷ √b = √(a ÷ b)
Same idea, but for division.
Worked Example
√21 ÷ √7 = √(21 ÷ 7) = √3
Move 3 — Factorising surds
√35 = √(5 × 7) = √5 × √7
This is just Move 1 run backwards — splitting one surd into two.
This "reverse" move is actually the most important one in the whole chapter, because it's the engine behind simplifying surds (Topic 3).
Move 4 — Adding and subtracting surds (the tricky one)
Here's where people go wrong. Adding and subtracting surds works exactly like collecting like terms in algebra — and nothing else. Think about how you'd handle 3x + 8x = 11x. You never touch what's inside the "x" — you just count how many of them you have. Surds behave the same way, treating √5 like a variable:
3√5 + 8√5 = 11√5 | 7√3 − 4√3 = 3√3
Only combine surds with the SAME number under the root.
2√3 + 4√6cannot be simplified any further — √3 and √6 are different "types" of surd, just like you can't combine 2x + 4y into a single term.
The #1 Mistake in This Whole Chapter
You can multiply and divide numbers under the square root sign. You absolutely cannot add or subtract them under there. Watch this:
√9 + √4 = 3 + 2 = 5 ✓ (do the square roots first, then add)
The rule √a + √b = √(a+b) is simply false. It only ever works for multiplication and division, never addition or subtraction.
Examiner Tip
If your calculator gives you an answer as a surd, keep it as a surd all the way through your working — don't convert it to a rounded decimal partway through. Rounding early is one of the sneakiest ways marks get lost, because tiny rounding errors snowball through multi-step questions.
Practice Question
Simplify, if possible: (a) 5√7 + 2√7 (b) √8 + √5
▶ Show Answer
3 · Simplifying Surds
This is where Move 3 (factorising) becomes genuinely useful. Simplifying a surd means digging out any perfect square that's hiding as a factor inside the number, and "rescuing" it from under the root sign.
The method
Find a perfect square (4, 9, 16, 25, 36...) that divides exactly into your number. If there's more than one option, always pick the largest one — it saves you having to simplify again afterwards.
Split the surd using √(ab) = √a × √b.
Take the square root of the perfect-square part — it becomes a normal number sitting outside the root sign.
√48 = √(16 × 3) = √16 × √3 = 4 × √3 = 4√3
16 was the largest perfect-square factor of 48, so it got "rescued" out.
Worked Example
√8 = √(4 × 2) = √4 × √2 = 2√2
√720 = √(144 × 5) = √144 × √5 = 12√5
Notice how in both cases we spotted the biggest perfect square factor first (4, then 144) rather than a smaller one — that way we simplify fully in one go.
How to spot the largest perfect-square factor fast
Run through the perfect squares in your head from biggest to smallest that could plausibly divide your number: 100, 81, 64, 49, 36, 25, 16, 9, 4. For 720, you might not spot 144 immediately — but you could also simplify in stages: √720 = √(4×180) = 2√180 = 2×√(36×5) = 2×6√5 = 12√5. Same answer, just via smaller steps. Either way works — being thorough beats being fast.
Simplifying and then combining
Often a question gives you two surds that look totally unrelated — until you simplify each one separately, and suddenly they turn out to be "like" surds that can be collected together.
Worked Example
√32 + √8
Rewrite each separately: √32 = √(16×2) = 4√2 and √8 = √(4×2) = 2√2
Now they're like surds: 4√2 + 2√2 = 6√2
Expanding brackets containing surds
Sometimes you'll need to multiply out double brackets that contain surds — this works exactly like expanding algebraic brackets (FOIL), with one extra trick: (√a)² = a. Whenever you multiply a surd by itself, it becomes a whole number and escapes the root sign entirely.
Worked Example
(√6 − 2)(√6 + 4)
Expand term by term: (√6)² + 4√6 − 2√6 − 8
Simplify: 6 + 2√6 − 8 = −2 + 2√6
Practice Question — from the source material
Write √54 − √24 in the form √q, where q is a positive integer.
▶ Show Answer
Practice Question
Expand and simplify: (√5 + 3)(√5 − 1)
▶ Show Answer
4 · Rationalising Denominators
Imagine trying to measure something with a ruler marked in an irrational unit — it just doesn't feel "usable." Mathematicians feel the same way about having a surd sitting on the bottom of a fraction, like 4/√5. It's not wrong, but it's considered messy and un-simplified. Rationalising the denominator is the process of rewriting the fraction so the bottom is a whole, rational number instead — even if the top ends up with a surd on it.
The key trick behind everything in this section
Every method here relies on one idea: √b × √b = b. Multiplying a surd by itself turns it into a whole number. So if you multiply top and bottom of a fraction by the right thing, you can force the surd on the bottom to "cancel out" into a whole number — while the fraction's actual value never changes, because multiplying by (something ÷ itself) is really just multiplying by 1.
Case A — Simple denominator (just a surd)
If the denominator is a single surd, √b, multiply top and bottom by √b.
a/√b = (a/√b) × (√b/√b) = (a√b)/b
Multiplying by √b/√b = 1, so the value doesn't change — but now the bottom is rational.
Worked Example
4/√5 = (4/√5) × (√5/√5) = (4√5)/5
The bottom went from an irrational √5 to a clean, rational 5. Job done.
Case B — Harder denominator (an expression with a surd)
This is where it gets more interesting. If the denominator is something like 3 + √5, multiplying top and bottom by √5 alone won't clean things up — you'd still have a surd floating around after expanding. Instead, you multiply by the conjugate: the exact same expression but with the sign in the middle flipped.
For denominator (a + √b), multiply top and bottom by (a − √b)
This pairing is called the "conjugate" — same numbers, opposite sign.
Why does flipping the sign help? Because of the difference of two squares:
(a + √b)(a − √b) = a² − (√b)² = a² − b
The surd terms cancel out completely, leaving a whole number. This is what makes the denominator rational.
Worked Example
Rationalise: 2/(3 + √5)
Step 1 — Multiply top and bottom by the conjugate (3 − √5):
2/(3+√5) = [2/(3+√5)] × [(3−√5)/(3−√5)]
Step 2 — Multiply out top and bottom:
= 2(3−√5) / [(3+√5)(3−√5)] = (6 − 2√5) / (9 − 5)
Step 3 — Simplify:
= (6 − 2√5) / 4 = (3 − √5) / 2
Examiner Tip
If, after all your working, your final answer still has a surd sitting on the bottom of the fraction — stop. Something has gone wrong somewhere, and it's worth going back to check your working, because a genuinely rationalised answer will never have a surd left in the denominator.
Practice Question — from the source material
Write 4/(√6 − 2) in the form p + q√r, where p, q and r are integers and r has no square factors.
▶ Show Answer
Practice Question
Rationalise the denominator: 6/√3
▶ Show Answer
What to Memorise
Rule
Statement
Definition
A surd is √(non-square integer) — irrational, cannot be simplified to a whole number.
Multiplying
√a × √b = √(ab)
Dividing
√a ÷ √b = √(a ÷ b)
Adding/Subtracting
Only combine LIKE surds (same number under root) — like collecting algebraic terms. NEVER √a + √b = √(a+b).
Simplifying
√(ab) = √a × √b — pull out the LARGEST perfect-square factor first.
Squaring a surd
(√a)² = a — this is the key trick for expanding brackets and rationalising.
Rationalise (simple)
Multiply top and bottom by the surd on the denominator: a/√b → (a√b)/b
Rationalise (harder)
Multiply top and bottom by the conjugate (flip the middle sign): (a+√b) → (a−√b)
Difference of two squares
(a + √b)(a − √b) = a² − b — this is what cancels the surd from the denominator.
Concepts Checklist
Exam Tips & Common Mistakes
Trap: Writing √a + √b = √(a+b). This is the single most common error examiners see. Addition and subtraction NEVER combine under the root sign — only multiplication and division do.
Trap: Not picking the largest perfect-square factor when simplifying, e.g. writing √48 = √4 × √12 = 2√12 and stopping there. Always check if the remaining surd (√12) can be simplified further — in this case it can (into 2√3), giving the fully simplified 4√3.
Trap: Rationalising a two-term denominator by multiplying by the surd only, instead of the full conjugate. Multiplying 2/(3+√5) by just √5/√5 will NOT remove the surd — you must use the conjugate (3−√5).
Trap: Rounding a surd to a decimal partway through a multi-step calculation. This introduces error that compounds through the rest of the working and can cost accuracy marks even if your method is otherwise correct.
What examiners reward: Clear step-by-step working — showing the factorisation used to simplify a surd, or the conjugate used to rationalise — even if you don't reach the final answer. Method marks are often available.
Exam pattern to expect: "Write [expression] in the form p + q√r" or "in the form √q" — these are asking you to fully simplify and match a given format exactly, so always double-check your final answer is in the simplest possible surd form before matching it to p, q, r.
Quick self-check: After rationalising, ask "is there still a surd on the bottom?" If yes, you've made an error — go back and check your conjugate or multiplication.