Surds

A surd is a root that cannot be written exactly as a fraction or a terminating decimal — so we leave it in root form to keep the answer exact. Surds are the reason questions say “give your answer in exact form” or “in the form a + b√2”.

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1. What a surd is

A surd is an irrational root — a root that cannot be simplified to a rational number.

Expression	Surd?	Why
√2	Yes	1.41421… never terminates or repeats
√3, √5, √6	Yes	irrational
√4	No	= 2, a whole number
√9	No	= 3
√12	Yes	but it simplifies to 2√3

A root of a square number is not a surd. √1, √4, √9, √16, √25 all give whole numbers. Recognising which roots come out exactly is the foundation of the whole topic — so know your square numbers to 20².

π is irrational but is not a surd, because it isn’t a root.

Rational vs irrational: a rational number can be written as a fraction — including integers, terminating decimals and recurring decimals. So 0.666… is rational (it is 2/3), a point specifically got wrong in lessons. Irrational numbers, such as √2 and π, cannot be written as any fraction.

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2. Simplifying surds

√(ab) = √a × √b

Look for a factor that is a square number.

Simplify √12:

1. 12 = 4 × 3 (and 4 is a square number)
2. √12 = √4 × √3
3. = 2√3

More examples:

• √18 = √9 × √2 = 3√2
• √50 = √25 × √2 = 5√2
• √72 = √36 × √2 = 6√2
• √48 = √16 × √3 = 4√3

Use the LARGEST square factor. √72 = √4 × √18 = 2√18 is correct but not fully simplified — √18 still contains a square factor. Using 36 gets you there in one step.

12 = 4 × 3, not 4². Confusing “a square factor” with “squaring” was a recorded error — you are looking for a factor that is a square number, then taking its root outside.

If you can’t spot the factor, use prime factorisation: 72 = 2³ × 3² → pairs of primes come out of the root → 2 × 3 = 6 outside, with a 2 left inside → 6√2.

5√2 does not mean 5 ÷ 2. It means 5 × √2. Reading it as a fraction was a documented error.

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3. Adding and subtracting

You can only add or subtract surds with the SAME number under the root — “like surds”, exactly like like terms in algebra.

• 3√2 + 5√2 = 8√2 ✓
• 7√3 − 2√3 = 5√3 ✓
• √2 + √3 — cannot be simplified ✓
• √2 + √2 = 2√2, not √4 and not 2

Never add the numbers under the roots. √2 + √3 is not √5. Check it: √2 + √3 ≈ 1.414 + 1.732 = 3.146, but √5 ≈ 2.236. This is the most common error in the topic.

Simplify first, then combine. Terms that look unlike often aren’t:

√8 + √18 = 2√2 + 3√2 = 5√2

Always simplify each surd before deciding whether they’re like. Concluding “these can’t be added” without simplifying loses easy marks.

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4. Multiplying and dividing

√a × √b = √(ab) and √a ÷ √b = √(a/b)

• √3 × √5 = √15
• √2 × √8 = √16 = 4
• √12 ÷ √3 = √4 = 2
• √a × √a = a — so √7 × √7 = 7

With coefficients, multiply the outside numbers and the surds separately:

2√3 × 4√5 = (2 × 4)(√3 × √5) = 8√15

Expanding brackets — treat surds like algebra:

√2(3 + √2) = 3√2 + √2×√2 = 3√2 + 2

(2 + √3)(4 + √3)

• 8 + 2√3 + 4√3 + √3×√3
• = 8 + 6√3 + 3
• = 11 + 6√3

Do all four products, and collect the like surds. Getting “14 + 2√3” where “14 + 7√3” was correct came from missing one of the cross terms.

√3 × √3 = 3, not √9 left unsimplified and not 3√3.

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5. Rationalising the denominator

A surd should not be left in the denominator.

Simple case — a single surd

Multiply top and bottom by that surd:

3/√2 = (3 × √2)/(√2 × √2) = 3√2 / 2

Example: 6/√3 = 6√3/3 = 2√3

Multiply the top AND the bottom — you are multiplying by √2/√2, which equals 1, so the value is unchanged.

Harder case — a two-term denominator

Multiply by the conjugate: the same expression with the sign in the middle reversed.

Rationalise 5/(3 + √2):

1. Conjugate of 3 + √2 is 3 − √2
2. Multiply top and bottom: 5(3 − √2) / [(3 + √2)(3 − √2)]
3. Denominator: 9 − 3√2 + 3√2 − 2 = 9 − 2 = 7 (the surd terms cancel)
4. = (15 − 5√2) / 7

The conjugate works because it creates a difference of two squares: (a + b)(a − b) = a² − b², and squaring a surd removes the root. That is the whole reason it eliminates the surd.

Change only the sign in the middle. Using 3 + √2 again, or negating both terms, leaves a surd in the denominator.

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6. Exam requirements

“Exact form” means leave the surd in. Writing √5 as 2.24 when the question said “exact” throws the mark away — this instruction appears constantly.

“In the form a + b√2” tells you what the answer must look like. Give the values of a and b, fully simplified.

Show your working. Tutors were explicit: on surd questions you do not get full marks for the answer alone.

Surds can appear on calculator papers too. A calculator will not give you an exact simplified surd, so the manipulation has to be done by hand either way.

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7. Mistakes that cost marks

Adding the numbers under the roots — √2 + √3 ≠ √5.

Not simplifying first, then wrongly concluding surds can’t be combined.

Not using the largest square factor.

Reading 5√2 as a fraction.

Leaving √3 × √3 as √9.

Missing cross terms when expanding brackets.

Leaving a surd in the denominator.

Multiplying only the numerator when rationalising.

Using the wrong conjugate.

Giving a decimal when an exact answer was required.

Calling √4 a surd, or 0.666… irrational.

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Frequently asked questions

What is a surd? An irrational root — one that can’t be written exactly as a fraction or terminating decimal, like √2.

Is √4 a surd? No — it equals 2.

How do I simplify a surd? Find the largest square factor, split the root, and take the square root of that factor outside.

Can I add √2 and √3? No — only surds with the same number under the root can be combined.

What is √2 + √2? 2√2.

What is √5 × √5? 5.

How do I multiply surds? √a × √b = √(ab); multiply any coefficients separately.

What does rationalising the denominator mean? Removing the surd from the bottom by multiplying top and bottom by the surd, or by the conjugate.

What is a conjugate? The same two-term expression with the middle sign reversed — the conjugate of 3 + √2 is 3 − √2.

What does “exact form” mean? Leave the surd in — don’t convert to a decimal.

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Quick revision checklist

• I know what a surd is and can spot roots that aren’t surds
• I know my square numbers to 20²
• I know recurring decimals are rational
• I can simplify a surd using the largest square factor
• I can use prime factorisation when the factor isn’t obvious
• I only add or subtract like surds
• I never add the numbers under the roots
• I simplify before deciding whether surds are alike
• I can multiply and divide surds, including coefficients
• I know √a × √a = a
• I can expand brackets with surds, collecting like terms
• I can rationalise a single-surd denominator
• I can use the conjugate for a two-term denominator
• I leave answers in exact form when asked
• I show my working

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These notes cover surds in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.