Sec 3 A-Math Surds: Rationalising & Solving

Overview

What you’ll learn

Decide whether a root is a surd and explain why surds give exact values.
Simplify square roots by extracting perfect-square factors and collect like surds.
Multiply and divide expressions with surds, including expanding brackets with conjugate pairs.
Rationalise denominators of the form 1/√a and a/(b + c√d) using the conjugate.
Solve equations involving surds by squaring, and reject extraneous solutions by checking.

In 30 seconds

1A surd is an irrational root kept in exact form, like √2 or 5√3 — the decimal never ends or repeats.
2Simplify with perfect-square factors: √ab = √a × √b, so √72 = √36 × √2 = 6√2.
3Only like surds add: a√c + b√c = (a + b)√c. The expression √2 + √3 cannot be combined.
4Multiplication and division work under the root: √a × √b = √(ab) and √a ÷ √b = √(a/b).
5Rationalise a denominator by multiplying top and bottom by the conjugate: 1/(a + √b) × (a − √b)/(a − √b).
6The conjugate of a + √b is a − √b — their product is the difference of two squares, a² − b.
7Squaring a surd equation can create fake solutions: every candidate must go back into the original equation.

Notes · 1 of 5

What surds are, and why they matter

A surd keeps an irrational value exact — the calculator decimal is only an approximation.

Some square roots of whole numbers are whole themselves: √4 = 2, √9 = 3, √25 = 5. Most are not: √2, √3, √5, √10 have decimals that run forever without repeating. A root like this, left in exact form, is a surd. Writing √2 ≈ 1.4142135 loses information; √2 is exact.

Exact or not?
Expression	Value	Surd?
√16	4 (a whole number)	No
√2	Irrational, exact as √2	Yes
5√3	Irrational, exact	Yes
1.414	A terminating decimal (≈ √2 but not equal)	No — an approximation

Surds obey the same field rules as other numbers: they can be added (when alike), multiplied, divided and bracketed. The rest of the chapter is the mechanics of doing that cleanly.

Notes · 2 of 5

Simplifying surds and collecting like terms

Pull out perfect-square factors, then add only surds of the same kind.

The rule √ab = √a × √b (for positive a and b) lets you extract perfect-square factors. Look for 4, 9, 16, 25, 36, 49, … hiding inside the number.

1Find a perfect-square factor
√72: 72 = 36 × 2, and 36 is a perfect square.
2Split the root
√72 = √36 × √2.
3Take the root of the square
√72 = 6√2. This is the simplest form — 2 has no square factor left.

Unlike surds do not merge: √2 + √3 stays as √2 + √3. There is no rule turning a sum of roots into the root of the sum — √2 + √3 ≈ 3.146, but √5 ≈ 2.236, so they are different numbers.

Notes · 3 of 5

Multiplying and dividing surds

Multiply under the root, expand brackets term by term, and watch for the difference of two squares.

The same rule runs in both directions: √a × √b = √(ab) and √a ÷ √b = √(a/b). So √8 × √2 = √16 = 4, and √27 ÷ √3 = √9 = 3 — sometimes a product of surds is not a surd at all.

1Expand brackets term by term
(2 + √3)(√3 + 1) = 2√3 + 2 + 3 + √3 — four products, just like ordinary algebra.
2Square roots multiply to whole numbers
√3 × √3 = 3: replace each pair with its number.
3Collect the like surds
2√3 + √3 = 3√3, so the product is 5 + 3√3.

Division sometimes collapses without any conjugate: (√8 − √2) ÷ √2 = (√8 ÷ √2) − (√2 ÷ √2) = √4 − 1 = 1. Try the term-by-term split before reaching for a more complicated method.

Notes · 4 of 5

Rationalising the denominator

Multiply top and bottom by the conjugate so the denominator becomes an ordinary integer.

A fraction is regarded as simplified when its denominator is rational. Two cases cover everything in the syllabus:

The two rationalising cases
Form	Multiply by	Why it works
1/√a (single surd below)	√a/√a	√a × √a = a, an integer
1/(a + √b) (sum or difference)	the conjugate (a − √b)/(a − √b)	(a + √b)(a − √b) = a² − b

1Identify the denominator
Is it one surd, or a sum with a surd?
2Multiply top and bottom
Whatever kills the denominator must multiply both the numerator and the denominator.
3Tidy the denominator
It becomes an integer: a, or a² − b.
4Simplify the fraction
Cancel common factors; write the result in the form the question asks for.

After rationalising, always look for a cancellation. In 10/(2√5), cancelling the factor of 2 first gives 5/√5, which rationalises to 5√5/5 = √5 — stopping at the uncancelled 10√5/10 (or mis-cancelling to √5/2) loses the final mark.

Notes · 5 of 5

Solving equations involving surds

Isolate the root, square both sides, solve the quadratic — then check every candidate.

An equation like √(2x + 3) = x is solved by squaring both sides, because squaring undoes the square root. But squaring is not reversible for negative values: two different numbers, x and −x, share the same square. That is why every solution must be checked in the original equation.

1Isolate the surd
Get √(expression) alone on one side before squaring.
2Square both sides
(√A)² = A, and the other side squares in full — including any coefficient.
3Solve the quadratic
Factorise, complete the square or use the formula.
4Check each candidate
Substitute into the original equation. A left side that is a root can never equal a negative right side — such candidates are rejected.

Revise

Key terms

The words examiners expect you to use precisely.

Surd: An irrational root left in exact form, such as √2 or 5√3, rather than written as a decimal.
Rational number: A number expressible as a fraction of integers, with a terminating or repeating decimal.
Irrational number: A number that is not a fraction of integers; its decimal never ends or repeats.
Simplest form: A surd with no perfect-square factor under the root, e.g. 6√2 rather than √72.
Like surds: Multiples of the same surd, such as 5√3 and −2√3; only like surds add or subtract.
Conjugate pair: Two expressions a + √b and a − √b; their product is the integer a² − b.

Show 4 more key terms

Rationalising the denominator: Multiplying a fraction by 1 (in conjugate form) so the denominator becomes rational.
Extraneous solution: A candidate produced by squaring that fails the check in the original equation.
Exact form: An answer written with surds (and/or fractions) so no rounding has occurred.
Principal square root: The non-negative root chosen by the √ symbol: √4 = 2, not −2.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Believing √(a + b) = √a + √b, e.g. claiming √9 + √16 = √25.
The root of a sum is not the sum of the roots: √9 + √16 = 3 + 4 = 7, but √25 = 5.
Instead: Multiplication splits and joins under the root; addition does not.
Mistake: Adding unlike surds, e.g. writing √2 + √3 = √5.
√2 and √3 are different irrationals; nothing lets them merge.
Instead: Convert each root to simplest form first; only multiples of the same surd combine.

Show 8 more mistakes

Mistake: Forgetting to square the coefficient: (2√3)² = 6.
(2√3)² = 2² × 3 = 12; the 2 is inside the square.
Instead: Square every factor: (k√a)² = k²a.
Mistake: Multiplying only the denominator by the conjugate when rationalising.
That changes the value of the fraction; the numerator must be multiplied by the same thing.
Instead: Multiply by conjugate/conjugate — a disguised 1 — so the value is preserved.
Mistake: Using the wrong conjugate sign, e.g. multiplying 1/(3 + √2) by (√2 + 3)/(√2 + 3).
Repeating the same sign gives (3 + √2)², which still contains surds.
Instead: The conjugate flips the sign: (3 + √2)(3 − √2) = 9 − 2 = 7.
Mistake: Accepting both quadratic candidates in a surd equation, e.g. answering x = 3 or x = −1 for √(2x + 3) = x.
Squaring is not reversible: −1 fails the original equation because √1 = 1 ≠ −1.
Instead: Check every candidate in the original equation and reject the failures.
Mistake: Writing √4 = ±2.
The √ symbol names the principal (non-negative) root only.
Instead: √4 = 2. The equation x² = 4 has solutions x = ±2, but that is a different statement.
Mistake: Leaving answers uncancelled, e.g. stopping at 10√5/10 instead of √5 when rationalising 10/(2√5).
The integer factors of numerator and denominator must cancel.
Instead: Treat the rational part as a fraction and simplify it; surds divide by their coefficient.
Mistake: Expanding (√5 + 3)(√5 − 4) and writing √5 × √5 = 10.
√5 × √5 = 5, not 10.
Instead: A surd times itself is the number under the root: (√k)² = k.
Mistake: Giving a rounded decimal when the question asks for an exact value, e.g. 2√2 → "2.83".
A rounded decimal is not exact and loses the answer mark in "without a calculator" or "exact form" questions.
Instead: Keep surds symbolic; round only when a question explicitly asks for 3 significant figures.

Check yourself

Quick quiz

18 questions with instant explanations. About 5 minutes.

Question 1 of 3

Question 1: current question
Question 2: not answered
Question 3: not answered

Recognise surds

Which of these is a surd?

Practise

Topical questions

Exam-style practice in three tiers, from recall to data questions.

5Foundation
5Standard
5Challenging

15 questions worth 60 marks, each with a model answer, mark scheme and examiner tips.

Practise topical questions

Exam practice

Practice paper

A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.

Question paper

Chapter 2 Practice Paper

30 marks · 40 min · 4 pages

5 multiple-choice questions and structured questions with answer lines. Chapter 2 (A3 Surds).

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Mark scheme

PRO

Chapter 2 Practice Paper — mark scheme

Answers · Marking points · 3 pages

Model answers with a mark for each creditworthy point — for marking at home or in class.

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Where this chapter is tested

Each school sets its own scope — check your assessment outline.

Questions

Frequently asked questions

What does Sec 3 A-Math Chapter 2 cover?

It covers syllabus sub-topic A3 of O-Level 4049 / SEC G3 K341 in full: simplifying surds, the four operations on surds, rationalising denominators including conjugate pairs, and solving equations involving surds with checks for extraneous solutions.

Why do I have to rationalise denominators?

A rational denominator is the standard "simplest form" convention in examinations: it makes two answers easy to compare and keeps further algebra clean. Questions explicitly ask for answers "with a rational denominator", and marks depend on it.

Why does a surd equation have a fake solution?

Squaring both sides is not reversible: x and −x have the same square, so the squared equation can hold where the original one does not. That is why every candidate goes back into the original equation — the ones that fail are called extraneous.

When do I need exact surd answers instead of decimals?

Whenever the question says "exact form", "give your answer in surd form" or "without a calculator". Decimals such as 1.414 are approximations of √2, not equal to it, and they lose the answer mark in those questions.

When is this chapter usually tested in school?

Surds sit early in the Sec 3 year, so they typically appear in Weighted Assessment 1 (Term 1) and again in the end-of-year examination. Scope and weighting vary by school, so check your school’s assessment outline.

Sec 3 A-Math Surds notes