The idea

The index laws

Why multiplying powers adds the indices, dividing subtracts them, and a power of a power multiplies them — and why every part of a bracket, coefficient included, takes the power.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

Number · Indices

The index laws

Why multiplying powers adds the indices, dividing subtracts them, and a power of a power multiplies them — and why every part of a bracket, coefficient included, takes the power.

Why it works

Count the factors

A power is just repeated multiplication:

2^3

means

2 \times 2 \times 2

. Every index law falls out of counting the factors — none of them needs memorising once you can see the count:

a^m \times a^n = a^{m+n}, \qquad \frac{a^m}{a^n} = a^{m-n}, \qquad (a^m)^n = a^{mn}

Multiplying powers adds the indices

2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^{3+4} = 2^7.

Three 2s times four 2s is seven 2s — the indices add. They do not multiply (

2^{12}

is wrong), and the base does not change (

4^7

is wrong): you still only ever multiply 2s.

Keep reading — free

The rest of the explanation, plus 2 worked examples you step through move by move.

Start free

Takes a minute — no card.