Index laws (also called laws of indices or exponent rules) tell you how to simplify expressions with powers. They appear throughout GCSE Maths—on both Foundation and Higher Tier papers for AQA, Edexcel, and OCR—and they underpin standard form, surds, and algebraic manipulation.

This guide lists every index law you need, shows how to combine them in harder questions, and finishes with two typical exam-style problems: writing a number as a power of 4, and writing a fraction as a power of 2.

What is an index?

In an expression like 2⁵, the 2 is the base and the 5 is the index (or exponent). The index tells you how many times to multiply the base by itself:

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

We read 2⁵ as “2 to the power 5” or “2 to the fifth”.

The index laws you must know

These rules work for any positive base (and for most laws when the base is a variable, provided you do not ÷ by zero or take even roots of negatives).

1. Multiplying powers (same base)

a^m × aⁿ = a^m+n

When the bases match, add the indices.

Example: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128

2. Dividing powers (same base)

a^m ÷ aⁿ = a^m−n

Subtract the indices.

Example: 5⁷ ÷ 5² = 5⁷⁻² = 5⁵

3. Power of a power

(a^m)ⁿ = a^mn

Multiply the indices.

Example: (3²)⁴ = 3^2×4 = 3⁸

4. Power of a product

(ab)ⁿ = aⁿbⁿ

Example: (2 × 5)³ = 2³ × 5³ = 8 × 125 = 1000

5. Power of a quotient

(ab)ⁿ = aⁿ/bⁿ

Example: (34)² = 3²/4² = 916

6. Zero index

a⁰ = 1 (for a ≠ 0)

Example: 7⁰ = 1, and (−4)⁰ = 1

7. Negative index

a⁻ⁿ = 1aⁿ

A negative index means “one over” the positive power.

Example: 2⁻³ = 12³ = 18

8. Fractional index

a¹ⁿ = ⁿ√a (the nth root of a)

a^mn = (ⁿ√a)^m

Example: 8¹³ = 2 because 2³ = 8

Example: 16³⁴ = (16¹⁴)³ = 2³ = 8

Quick reference table

Multiply, same base: add indices — a^m × aⁿ = a^m+n
Divide, same base: subtract indices — a^m ÷ aⁿ = a^m−n
Power of a power: multiply indices — (a^m)ⁿ = a^mn
Product raised to a power: (ab)ⁿ = aⁿbⁿ
Quotient raised to a power: (ab)ⁿ = aⁿ/bⁿ
Zero index: a⁰ = 1
Negative index: a⁻ⁿ = 1aⁿ
Fractional index: a¹ⁿ = ⁿ√a

Combining index laws

Exam questions often need more than one law. Work step by step: simplify brackets first (power of a power), then multiply or ÷.

Example: Simplify (2³)² × 2⁴ ÷ 2⁵

Step 1 — power of a power: (2³)² = 2⁶

Step 2 — multiply: 2⁶ × 2⁴ = 2¹⁰

Step 3 — ÷: 2¹⁰ ÷ 2⁵ = 2⁵ = 32

Example: Simplify (x²y³)⁴ ÷ (x³y)²

Step 1 — expand brackets: (x²)⁴(y³)⁴ ÷ (x³)²(y)² = x⁸y¹² ÷ x⁶y²

Step 2 — ÷ (subtract indices): x⁸⁻⁶y¹²⁻² = x²y¹⁰

Example: Write 81⁻¹² as a single number

81⁻¹² = 181¹² = 19 = 19

Writing numbers in index form (exam style)

Many GCSE questions ask you to write a number using a given base. The strategy is always the same:

Write the number as a power of a prime (usually 2, 3, or 5).
Rewrite the required base as a power of that same prime.
Use (a^m)ⁿ = a^mn to match the index.

Exam question (a): Write 32 in the form 4ⁿ

Question: Write 32 in the form 4ⁿ.

Solution:

Write 32 as a power of 2: 32 = 2⁵

Write 4 as a power of 2: 4 = 2²

So 4ⁿ = (2²)ⁿ = 2²ⁿ

Set equal to 32:

2²ⁿ = 2⁵

Same base, so 2n = 5, hence n = 52

Answer: 32 = 4⁵² (or n = 52)

Check: 4⁵² = (2²)⁵² = 2⁵ = 32 ✓

Exam question (b): Write 18 in the form 2²ⁿ

Question: Write 18 in the form 2²ⁿ.

Solution:

Write 18 using a negative index. Since 8 = 2³:

18 = 2⁻³

We want 2²ⁿ = 2⁻³

So 2n = −3, hence n = −32

Answer: 18 = 2²ⁿ when n = −32

Check: 2^2×(−32) = 2⁻³ = 18 ✓

Common mistakes to avoid

Adding bases: 2³ + 2⁴ is not 2⁷. Index laws apply to multiplication and division of powers with the same base, not addition.
Multiplying indices when you should add: 2³ × 2⁴ = 2⁷, not 2¹².
Forgetting brackets: (2³)² = 2⁶, but 2^3² would mean 2⁹ if misread—always use brackets for “power of a power”.
Negative indices: 2⁻³ = 18, not −8.
Fractional answers: Questions like part (a) often have a fractional index—leave your answer as 52 unless asked for a decimal.

Practice questions

Try these before reading the answers.

Simplify 3⁴ × 3² ÷ 3³
Simplify (5²)³
Write 16 in the form 2ⁿ
Write 127 in the form 3ⁿ

Click to reveal answers

1. 3⁴⁺²⁻³ = 3³ = 27

2. 5⁶ = 15 625

3. 16 = 2⁴, so n = 4

4. 127 = 3⁻³, so n = −3

Summary

Index laws let you simplify powers quickly: add indices when multiplying (same base), subtract when dividing, and multiply when raising a power to another power. Combine them carefully for multi-step questions, and use prime factorisation when an exam paper asks you to write a number in a specific index form—such as 32 = 4⁵² or 18 = 2⁻³ = 2²ⁿ with n = −32.

Master these rules and you will find standard form, growth and decay, and algebraic simplification much easier. Good luck with your revision!

Index Laws: A Complete GCSE Maths Guide with Examples