Laws of indices

Index laws are shortcuts for working with powers of the same number. Multiplying adds the powers, dividing subtracts them, and a power of a power multiplies them. Zero, negative and fractional powers follow from the same rules.

Key facts

  • aᵐ × aⁿ = aᵐ⁺ⁿ
  • aᵐ ÷ aⁿ = aᵐ⁻ⁿ
  • (aᵐ)ⁿ = aᵐⁿ
  • a⁰ = 1, a⁻ⁿ = 1/aⁿ
  • a^(1/n) = ⁿ√a, a^(m/n) = (ⁿ√a)ᵐ

Worked examples

Example 1

Simplify x⁵ × x³ ÷ x².

  1. Multiplying: add the powers. x⁵ × x³ = x⁸
  2. Dividing: subtract. x⁸ ÷ x² = x⁶

Answer: x⁶

Example 2

Work out 8^(2/3).

  1. The bottom of the fraction is the root: cube root of 8. ∛8 = 2
  2. The top is the power. 2² = 4

Answer: 4

Example 3

Work out 5⁻².

  1. 5⁻² = 1/5²
  2. = 1/25

Answer: 1/25