IP EMaths Notes (Upper Sec, Year 3-4): 03) Algebraic Fractions and Surds

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

Simplify algebraic fractions, manage restricted domains, and keep surd expressions exact throughout working.

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For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.

How this chapter applies

Eclat core: factorising before cancellation and combining algebraic fractions are part of the Eclat foundation; surd manipulation is retained as an IP bridge.
School-sensitive extension: four-operation surd work and equations involving surds may be assessed as accelerated A-Math material rather than E-Math in some schools.
2027 national comparison: K310 Topic N5 includes multiplication, division, addition, and subtraction of algebraic fractions. Surds belong explicitly to K341 Topic A3, not the K310 core.
Check your school: check whether surds are assessed in the current E-Math paper, taught only as A-Math preparation, or postponed.
Exam-track route: use the separate O-Level and SEC G3 Mathematics notes for K310 topic ownership.

Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 03) Algebraic Fractions and Surds cover?
A: Simplify algebraic fractions, manage restricted domains, and keep surd expressions exact throughout working.

The core idea is simple: Factor first, cancel only common factors, and keep surds exact.

Use it as a working check: Algebraic fractions behave like number fractions, but restrictions come from the original denominator before cancellation. Surds stay cleaner when you rationalise with a matching conjugate.

Then go one layer deeper: Work through the examples to practise a fixed sequence: factor, state excluded values, simplify, rationalise if needed, and check that any squared surd equation still satisfies the original equation.

Exact manipulation avoids rounding drift. Reduce algebraic fractions fully before substituting values, and rationalise surd denominators when presenting final answers.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

Key techniques

Factor numerator and denominator before cancelling: $\dfrac{x^{2} - 9}{x^{2} - x - 6} = \dfrac{(x - 3)(x + 3)}{(x - 3)(x + 2)}$

Reviewed by

Marcus Pang·Managing Director (Maths)

Question cue	First move	Reason it works	Common trap
Single algebraic fraction to simplify	Factor numerator and denominator fully.	Cancellation is only valid for common factors, not common terms.	Cancelling the $x$ in $\dfrac{x + 3}{x + 5}$ .
Algebraic fractions to add or subtract	Find the lowest common denominator.	Every numerator must be rewritten over the same denominator before combining.	Combining the denominators directly as if they were like terms.
Denominator contains one surd	Multiply by that surd.	The denominator becomes a rational number because $(\sqrt{a})^2 = a$ .	Changing the value by multiplying only the denominator.
Denominator contains $a + \sqrt{b}$ or $a - \sqrt{b}$
Equation contains square roots	State the domain, isolate one surd, then square.	Squaring can introduce extra roots, so the final check is part of the solution.	Accepting a value that fails in the original equation.

Expression pattern	First rewrite	What to look for	Common trap
$\sqrt{18x}$ and $\sqrt{2x}$ appear together	$\sqrt{18x}=3\sqrt{2x}$	Same surd factor in numerator and denominator	Rationalising before checking for a common factor
$\sqrt{8x}$ appears with $\sqrt{2x}$
Denominator becomes $\sqrt{2x}+2\sqrt{2x}$

IP EMaths Notes (Upper Sec, Year 3-4): 03) Algebraic Fractions and Surds

How this chapter applies

Key techniques

Sources

First-move checkpoint

Shared surd factor checkpoint

Worked example - Simplify and rationalise

Worked example - Combine algebraic fractions with different denominators

Worked example - Solve an equation involving surds

Practice Quiz

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