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

Study guide08 Nov 2025, 00:00 ZUpdated 30 Nov 2025

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

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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.

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These notes align with SEAB GCE O-Level Mathematics (4052) content used in IP programmes (exams from 2026).

Status: SEAB O-Level Mathematics 4052 syllabus (exams from 2026) checked 2025-11-30 - scope unchanged; remains the reference for these 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

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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