Laws of indices

GCSE · 1MA1 · N7, A4
International GCSE · 4MA1 · 2.1
About 15 minutes

A power is shorthand for repeated multiplication. The index laws are just shortcuts for counting how many times something is multiplied.

Multiplying: add the powers

2³ × 2² = (2 × 2 × 2) × (2 × 2) = 2⁵
So a^m × aⁿ = a^m+n. The base must be the same.

Dividing: subtract the powers

a^m ÷ aⁿ = a^m−n
x⁷ ÷ x³ = x⁴

Power of a power: multiply

(a^m)ⁿ = a^mn
(x²)³ = x² × x² × x² = x⁶

Zero and negative powers

Anything (except 0) to the power 0 is 1: 5⁰ = 1
A negative power means one over: a^-n = 1aⁿ, so 3^-2 = 19

Worked example

Simplify (x⁴ × x⁵) ÷ x³

Brackets first. Multiply, so add powers: x⁴ × x⁵ = x⁹
Divide, so subtract powers: x⁹ ÷ x³ = x⁶

Answer: x⁶

Key idea

Same base: multiply means add the powers, divide means subtract, bracket to a power means multiply. a⁰ = 1 and a^-n = 1aⁿ.

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