SPEC A1, A3 • ALGEBRA — NOTATION & SIMPLIFYING
Algebraic Notation & Simplifying
The language of algebra: terms, coefficients, like terms, and the laws of indices — the foundation every algebra question is built on.
What You'll Learn
Algebra has its own dictionary. Before you can expand, factorise, rearrange or solve, you need the vocabulary straight — and the notation rules that silently govern every algebraic expression.
This Lesson Covers
Vocabulary: term, coefficient, like terms, expression, equation, identity
Notation conventions: how algebra is written and what the symbols mean
Collecting like terms: the first simplification every exam expects
Laws of indices applied to algebra — including negative and fractional indices H
Common traps examiners deliberately set on this topic
Why it matters
Algebra is around 30% of Higher and 20% of Foundation. Every later lesson — expanding brackets, factorising, solving equations, simultaneous equations, proofs — begins with the vocabulary and simplification you'll learn here.
Terms & Coefficients
Term
A term is a single number, variable, or product of numbers and variables.Examples: 3x −5 2ab 7x²
Coefficient
The coefficient is the numerical factor of a term.In 7x², the coefficient is 7If a term has no visible number (like x or ab), the coefficient is an implied 1. If it has a minus sign (like −5x), the coefficient is −5.
Like Terms
Like Terms
Terms with identical variable parts — the same letters, each raised to the same power.
LIKE TERMS ✓
3x² and −5x²
Both have variable part x². Coefficients can differ.
NOT LIKE TERMS ✗
3x² and 3x
Different powers. x² is not the same as x.
LIKE TERMS ✓
2ab and −7ab
Identical variable part: ab.
NOT LIKE TERMS ✗
2ab and 2a²b
Different power on a.
Expression · Equation · Identity
Expression
A combination of terms with no equals sign.3x + 2y − 5
Equation
An expression equal to another. True for specific values of the variable.3x + 2 = 11 (true only when x = 3)
Identity
An equation true for every value of the variable. Uses the ≡ symbol (triple bar).(x + 1)² ≡ x² + 2x + 1Try any value of x — both sides always agree.
Three More Vocabulary Words
The AQA spec (A3) requires you to understand the vocabulary of these three as well — even though each gets its own dedicated lesson later.
Formula
A rule written as an equation, usually linking two or more quantities. Formulae are given — you substitute values in.A = πr² C = 2πr v = u + atLesson 2.06 covers substitution and using formulae in depth.
Inequality
Like an equation, but with <, >, ≤ or ≥ instead of =. True for a range of values.3x + 2 < 11 (true for all x < 3)Lesson 2.20 covers solving linear inequalities.
Factor
A term that divides another exactly. Just like 4 is a factor of 12, (x + 2) is a factor of (x + 2)(x − 5).x² + 5x = x(x + 5) — here x and (x + 5) are factorsLessons 2.03 and 2.04 cover factorising.
Notation Conventions
Mathematical notation silently drops multiplication signs and follows quiet conventions. These are the exact notation rules the AQA spec (A1) requires.
What's writtenWhat it meansNote
aba × bAlphabetical order preferred
3yy + y + y (and also 3 × y)Repeated addition OR multiplication — both valid
a²a × aNot 2a. Never.
a³a × a × aNot 3a.
a²ba × a × bOnly the a is squared
a/ba ÷ bTreat as a single fraction
(2x + 3)Brackets = groupingEvaluate what's inside first (BIDMAS)
x alone1 × x¹Coefficient 1 and power 1 are implied
Spec rule: coefficients as fractions
AQA expects coefficients to be written as fractions, not decimals. Write ½x or x/2, not 0.5x. Write ¾a, not 0.75a.
Brackets Matter: (−x)² vs −x²
One of the most costly single-mark errors in algebra — and it comes from how you read the minus sign.
(−x)² with brackets
(−x)² = x²
The bracket says square the whole thing: (−x) × (−x) = +x². Negative × negative = positive.
−x² without brackets
−x² = −(x × x)
Only the x is squared. The minus sits outside, so the answer is negative.
Exam trap
Substituting x = −3 into x² gives +9, not −9 (because (−3)² = 9). But substituting x = 3 into −x² gives −9. Watch the brackets.
Collecting Like Terms
The Method
Identify like terms — group terms with the same variable part.
Add (or subtract) the coefficients.
Leave unlike terms alone — they can't combine.
Quick Examples
3x + 5x = 8x7a − 2a = 5a3a² − 2a² = a² (not 1a² — the 1 is implied)
Do not combine
3x + 4x² cannot be simplified further. Different powers means not like terms. The answer is 3x + 4x².
01Worked Example
Simplify: 3x² + 4x − 2x² + x [2 marks]
Step 1 — Group like terms
(3x² − 2x²) + (4x + x)
Step 2 — Add the coefficients
= (3 − 2)x² + (4 + 1)x
Step 3 — Write the simplified form
= x² + 5x
AQA Mark Scheme
AQA typically splits this as B1 for the x² coefficient and B1 for the x coefficient. Get one right, get one mark. Show both groupings explicitly.
Laws of Indices — The Core Three
Indices for algebra work exactly like indices for numbers. Memorise these three and most problems fall out.
LAW 1
Multiplying — same base, add powers.
xᵃ × xᵇ = xᵃ⁺ᵇ
e.g. x⁴ × x³ = x⁷
LAW 2
Dividing — same base, subtract powers (top minus bottom).
xᵃ ÷ xᵇ = xᵃ⁻ᵇ
e.g. m⁸ ÷ m² = m⁶
LAW 3
Power of a power — multiply the powers.
(xᵃ)ᵇ = xᵃᵇ
e.g. (n²)⁴ = n⁸
02Worked Example — Division of Indices
Simplify fully: 15a⁷b² ÷ 5a³b [2 marks]
Step 1 — Separate coefficients from variables
(15 ÷ 5) × (a⁷ ÷ a³) × (b² ÷ b)
Step 2 — Apply Law 2 (subtract: top minus bottom)
= 3 × a⁷⁻³ × b²⁻¹
Step 3 — Evaluate
= 3 a⁴ b
Watch the hidden power of 1
The b on the bottom is really b¹. So b² ÷ b¹ = b²⁻¹ = b¹ = b. Never leave b⁰ or skip the step.
Zero, Negative & Fractional Indices
LAW 4
Power of zero — anything (non-zero) to the power 0 is 1.
x⁰ = 1
e.g. 7⁰ = 1, (5a)⁰ = 1
LAW 5
Negative power — take the reciprocal. A negative index flips, it does not make the answer negative.
x⁻ᵃ = 1 / xᵃ
e.g. 2⁻³ = 1/2³ = 1/8
LAW 6 HIGHER
Fractional power — the denominator is the root, the numerator is the power. Root first, then power keeps numbers manageable.
x^1/n = ⁿ√x x^m/n = (ⁿ√x)ᵐ
e.g. 8^2/3 = (³√8)² = 2² = 4
Negative ≠ negative
2⁻³ is NOT −8. A negative index means "one over". 2⁻³ = 1/8. The answer is still positive.
Power of a Product or Quotient
When a bracket is raised to a power, everything inside gets that power — including the coefficient.
LAW 7
Product — distribute the power to each factor.
(xy)ᵃ = xᵃ yᵃ
e.g. (2x)³ = 2³ x³ = 8x³
LAW 8
Quotient — top and bottom both raised.
(x / y)ᵃ = xᵃ / yᵃ
e.g. (3/4)² = 9/16
Multi-letter bracket
(4a³b)² means every part squared: 4² × (a³)² × b² = 16 a⁶ b². The 4 must be squared too.
The (2x)³ Trap
This is the most common index-law error flagged by AQA examiners on Paper 1: students cube the x and forget the coefficient.
WEAK STUDENT ✗(2x)³ = 2x³
Only cubed the x. The 2 was ignored. Zero marks.
STRONG STUDENT ✓(2x)³ = 8x³
Everything inside is cubed: 2³ = 8, x³ = x³. So 8x³.
Rule of thumb
When you see a power outside a bracket, mentally rewrite as a product first: (2x)³ = (2x)(2x)(2x). Now it's obvious both 2 and x are cubed.
MCE Framework: The Worked Example
Now bring it all together — vocabulary, notation, index laws — into one 3-mark question answered using the MCE framework.
Question. Simplify fully:
(4a³b)² × 3a²b⁵
[3 marks]
Method
Apply power of a product (Law 7) to the bracket, then multiply coefficients and add powers (Law 1) of the same base.
Calculate
Shown on the next slide — every line of working written down.
Evaluate
State the final simplified expression clearly as the answer.
MCE Worked Example — Solution
STEP 1 — Square the bracket (Law 7: everything inside)
(4a³b)² = 4² × (a³)² × b² = 16 a⁶ b²
STEP 2 — Multiply the coefficients
16 × 3 = 48
STEP 3 — Add the powers of each base (Law 1)
a⁶ × a² = a⁶⁺² = a⁸
b² × b⁵ = b²⁺⁵ = b⁷
STEP 4 — Evaluate: write the final answer
= 48 a⁸ b⁷
Mark breakdown (typical AQA)
For simplification questions like this, AQA awards B marks (single-step facts), not M/A marks. Typical split: B1 for 16 a⁶ b² (bracket squared correctly), B1 for the correct coefficient 48, B1 for the fully-correct final expression 48 a⁸ b⁷.
Examiner Alerts
Alert 1 — Forgetting to square the coefficient
Students write: (4a³b)² = 4a⁶b² — the 4² has been dropped.
Write instead: Everything inside the bracket is squared, including the coefficient. So 4² = 16 — it becomes 16 a⁶ b².
Alert 2 — Subtracting indices the wrong way
Students write: x⁸ ÷ x³ = x⁻⁵ — bottom minus top.
Write instead: Always top minus bottom. x⁸ ÷ x³ = x⁸⁻³ = x⁵.
Alert 3 — Multiplying powers instead of adding
Students write: x² × x³ = x⁶ — they multiplied the indices.
Write instead: Multiplication of same base → add indices. x² × x³ = x⁵. Only power of a power multiplies.
Practice Question
Simplify fully:
(3xy²)³ × 2x⁴y
[3 marks]
M
Method. Power-of-a-product on the bracket first (everything inside cubed), then multiply coefficients and add powers of the same base.
C
Calculate.
(3xy²)³ = 3³ × x³ × (y²)³ = 27 x³ y⁶
× 2x⁴y: 27 × 2 = 54. x³ × x⁴ = x⁷. y⁶ × y = y⁷.
E
Evaluate. Final answer: 54 x⁷ y⁷.
Lesson Summary
Vocabulary to know word-for-word (Spec A3)
Term / Coefficient / Like terms — building blocks
Expression (no =) / Equation (= for specific values) / Identity (≡ for all values)
Formula (rule — Lesson 2.06) / Inequality (<, >, ≤, ≥ — Lesson 2.20) / Factor (divides exactly — Lessons 2.03–2.04)
The 8 Laws of Indices
1. xᵃ × xᵇ = xᵃ⁺ᵇ2. xᵃ ÷ xᵇ = xᵃ⁻ᵇ
3. (xᵃ)ᵇ = xᵃᵇ4. x⁰ = 1 (x ≠ 0)
5. x⁻ᵃ = 1/xᵃ6. x^1/n = ⁿ√x H
7. (xy)ᵃ = xᵃ yᵃ8. (x/y)ᵃ = xᵃ/yᵃ
Top 3 traps to avoid
(2x)³ = 8x³ — cube the coefficient too
x⁸ ÷ x³ = x⁵ — subtract top minus bottom
x² × x³ = x⁵ — add indices, don't multiply them
MCE reminder
Every multi-step algebra question should follow M → C → E: state the method, show every line of calculation, then write the final simplified answer. AQA awards B marks for simplification (single-step facts) and M/A marks for methods leading to an accuracy answer — show all working either way.
Next: What You'll Learn