Year 8 Mathematics · Number and Algebra · 9m read

Integers and Index Laws

Ever wondered why mathematicians invented negative numbers, or why 2³ is shorthand for 2 × 2 × 2? Integers (positive, negative, and zero) and index laws—the rules for working with powers—are the building blocks you'll use constantly in algebra and beyond. Getting comfortable with these now means you'll tackle harder maths with confidence, from solving equations to understanding exponential growth in real life.

Learning objective: Operate with integers and apply the index laws for products, quotients and powers.

By the end of this lesson you can:

• I add, subtract, multiply and divide integers.
• I apply index notation and the three index laws.
• I evaluate expressions with negative and zero indices.

Indices are shorthand for repeated multiplication. Writing 2 × 2 × 2 × 2 × 2 is tiring; 2⁵ says the same thing. Index laws are shortcuts for combining these shorthand notes.

Integers and index laws

INTEGER OPERATIONS:
SAME signs → positive. (+)(+) = + ; (−)(−) = +
DIFFERENT signs → negative. (+)(−) = − ; (−)(+) = −
This rule applies to × and ÷.

Adding/subtracting: use number line.
5 − (−3) = 5 + 3 = 8 (double negative → add)
−4 + (−6) = −10

INDEX NOTATION:
aⁿ: base a raised to exponent n. 3⁴ = 3 × 3 × 3 × 3 = 81.

THREE INDEX LAWS:
PRODUCT: aᵐ × aⁿ = aᵐ⁺ⁿ (same base → ADD exponents)
QUOTIENT: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base → SUBTRACT exponents)
POWER: (aᵐ)ⁿ = aᵐⁿ (power of a power → MULTIPLY exponents)

ZERO INDEX: a⁰ = 1 (any non-zero base to power 0 = 1)
NEGATIVE INDEX: a⁻ⁿ = 1/aⁿ (e.g. 2⁻³ = 1/2³ = 1/8)

CAREFUL:
(−3)² = 9 (positive!) BUT −3² = −9 (applies power then negates)
Index laws only work with the SAME BASE.

Index law cheatsheet

• Product law: 3⁵ × 3² = 3⁷. ADD the indices when bases are the same.
• Quotient law: 7⁸ ÷ 7³ = 7⁵. SUBTRACT the indices when bases are the same.
• Power of a power: (2³)⁴ = 2¹². MULTIPLY the indices.

Index laws

• Multiply: a^m \times a^n = a^{m+n}
• Divide: a^m \div a^n = a^{m-n}
• Power of power: (a^m)^n = a^{mn}
• Zero index: a^0 = 1

Quick reference

| Law | Rule | Example |
|---|---|---|
| Multiply | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| Divide | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
| Zero index | a⁰ = 1 | 7⁰ = 1 |
| Negative × negative | Positive | (−3)(−4) = 12 |

Worked example (Medium)

Simplify: 3⁵ × 3² ÷ 3⁴

1. Product law. 3⁵ × 3² = 3⁷
2. Quotient law. 3⁷ ÷ 3⁴ = 3³
3. Evaluate. 3³ = 27

Answer: 27

Worked example (Hard)

Apply advanced understanding of integers and index laws to a challenging real-world scenario. Demonstrate systematic analysis using key concepts from this topic.

1. Analyse the context. Identify the key principles of integers and index laws that are relevant. Consider what specific knowledge is needed to address this scenario.
2. Apply specific knowledge. Use precise mathematical terminology to analyse the situation. Show how concepts interact and build upon each other.
3. Evaluate implications. Consider the broader significance. How does this analysis connect to other aspects of maths?
4. Form a conclusion. Synthesise your analysis into a well-reasoned conclusion supported by specific evidence and terminology.

Answer: Analysis demonstrates deep understanding of integers and index laws with specific terminology, systematic reasoning, and real-world application.

Common mistakes

• Providing surface-level responses without depth Students state facts about integers and index laws without explaining underlying mechanisms, reasoning, or significance. Fix: Always explain WHY and HOW, not just WHAT. In mathematics assessments, depth of explanation distinguishes high-achieving responses from basic recall.
• Not using precise mathematical terminology Students use everyday language when specific terms would demonstrate stronger understanding. Fix: Build and use a vocabulary of key terms for this topic. Precise terminology shows understanding and earns higher marks in mathematics assessments.

Integers and Index Laws — connecting to the bigger picture

Understanding integers and index laws extends beyond this single topic — it connects to broader themes across maths and other disciplines. Recognising these connections transforms isolated knowledge into an integrated framework that supports deeper analysis and creative problem-solving.

In professional practice, experts use conceptual frameworks — organised structures of related ideas — to approach new problems efficiently. They don't recall every fact; they activate connections between concepts that help them identify what's relevant. This is why the most effective study approach is not memorising more facts, but building richer connections between the facts you already know.

Active learning strategies that build these connections include: relating new concepts to personal experience, comparing ideas across different topics, applying knowledge to unfamiliar scenarios, and teaching concepts to others. Each of these forces you to process information more deeply than passive reading or highlighting. The goal is to develop the kind of flexible understanding that allows you to analyse any question about integers and index laws — even ones you haven't seen before — by drawing on a network of connected ideas.

Practice questions

1. Simplify: (4²)³

   1. 4⁵
   2. 4⁶
   3. 4⁸
   4. 16³

   (1 marks)

2. Evaluate: 3⁻² + 2⁰

   (2 marks)

Summary

• Add, subtract, multiply and divide integers.
• Apply index notation and the three index laws.
• Evaluate expressions with negative and zero indices.