Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Expressions & Index Laws

In one sentence. Collecting like terms and running the index laws on letters — the fluency layer every other algebra module assumes.

Algebra — one of 8 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with François Viète. Learn Expressions & Index Laws, teach it, then defend it against François Viète.

Where this sits in the specification

Specification 4MA1, 2.1 Use of symbols · 2.2 Algebraic manipulation. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 2.1 — Use of symbols
  • 2.2 — Algebraic manipulation

1 · Understand it

What it is

A letter stands for a number we haven't chosen yet. 'Like terms' share exactly the same letter part (3x and −2x; 4a²b and a²b) and only like terms can be collected. The index laws work on letters exactly as on numbers: xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ.

Why it works

3x + 2x = 5x because three of something plus two of the same thing is five of that thing — the x never needs to be known. 3x + 2y stays as it is for the same reason you can't add 3 apples and 2 oranges into 5 of anything. The index laws are just counting factors: a² × a³ has 2 + 3 copies of a multiplied together.

When a question is asking for it

Simplifying is never the whole question — it's the opening move of almost every algebra question. Fluency here decides whether harder questions (equations, quadratics, proofs) feel possible or exhausting.

How to hold it in your head

Read 4a²b as a single object — 'an a²b' — and count how many of them you have. For indices, don't recite laws: picture the copies. (2x³)⁴ is four bags each holding 2·x·x·x, so 2⁴ = 16 and twelve x's.

2 · The mistakes that cost marks

  • Collecting unlike terms: 3x + 2y ≠ 5xy. No shared letter part, no collecting.
  • x × x = 2x. Multiplying gives x² (two copies multiplied); 2x is x + x (two copies ADDED).
  • (2x³)⁴ = 2x¹² — the coefficient must be raised too: 2⁴x¹² = 16x¹².
  • Adding indices when dividing. Division SUBTRACTS: x⁶ ÷ x² = x⁴.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Simplify 4a²b × 3ab³.

  1. Numbers first: 4 × 3 = 12.
  2. a's: a² × a = a³ (add indices; a is a¹).
  3. b's: b × b³ = b⁴.

Answer: 12a³b⁴

Simplify (2x³)⁴.

  1. The power 4 hits everything in the bracket.
  2. 2⁴ = 16 and (x³)⁴ = x¹².

Answer: 16x¹²

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Simplify 5x + 3y − 2x + 7y.
  2. Grade 7 — Simplify 4a²b × 3ab³.
  3. Grade 8 — Simplify 15x⁶y² ÷ 3x²y.
  4. Grade 9 — Simplify (3x²y)³ × 2xy⁴.
  5. 9+ — Simplify (27x⁶)^(2/3).

Ask the person who made it

Inside Incandio you can put a question to Ada Lovelace, who answers knowing exactly which module you have just finished: “You saw that the Analytical Engine could work on symbols, not just numbers — and that this changed everything. Why is doing mathematics with letters instead of numbers such a powerful idea?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Expressions & Index Laws, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Expressions & Index Laws works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Expressions & Index Laws against François Viète — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 11 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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