The five laws
A standard method from Indices and scientific notation: the steps and worked examples.
Steps
| Law | Why, in one line |
|---|---|
| aᵖ × a^q = a^(p+q) | p copies of a, then q more copies |
| aᵖ ÷ a^q = a^(p−q) | the copies on the bottom cancel |
| (aᵖ)^q = a^(pq) | q groups of p copies |
| (ab)ᵖ = aᵖbᵖ | each copy of ab gives one a and one b |
| (a/b)ᵖ = aᵖ/bᵖ | the same idea for a fraction |
a⁰ = 1, because aᵖ ÷ aᵖ = 1. a⁻ⁿ = 1/aⁿ, because a⁰ ÷ aⁿ = 1/aⁿ.
- Write the terms with the same base where you can (8 = 2³).
- Pick the law for each operation: × adds indices, ÷ subtracts them, a power of a bracket multiplies them.
- Work from the inside of the brackets outwards, one law per line.
Worked examples
Example 1: Use the five laws to evaluate (2³)² × 2⁻⁴ ÷ 2⁰.
(2³)² = 2⁶ power of a power: multiply
2⁶ × 2⁻⁴ ÷ 2⁰ = 2^(6 − 4 − 0)
= 2² = 4For the whole topic, read Indices and scientific notation: key points, common mistakes and how the DSE tests it.