An index is shorthand for how many times a number has been multiplied by itself. Every law comes from writing the copies out and counting.
| Law | Why it is true | Example |
|---|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | Put two collections side by side and count | 2³ × 2² = 2⁵ = 32 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Cancel matching copies and count what is left | 2⁵ ÷ 2² = 2³ = 8 |
| (aᵐ)ⁿ = aᵐⁿ | n groups of m copies each, so multiply | (2³)² = 2⁶ = 64 |
| a⁰ = 1 | A number divided by itself | 2⁰ = 1 |
16, 8, 4, 2, and each step halves, so the next must be 1. The division law agrees: 2³ ÷ 2³ is 2⁰, and a number over itself is 1.
2³ means 2 × 2 × 2 = 8, not 6. The index counts copies; it is not a multiplier.
Negative indices
A negative index does not make the answer negative; it puts the base underneath. 2⁻² is one quarter.
2³ × 3² cannot become a single power, because the bases differ. Work each out: 8 × 9 = 72.
Standard form
45000 becomes 4.5 × 10⁴ and 0.00032 becomes 3.2 × 10⁻⁴. The index records how many places the point moved.
Big number, positive index. Small number, negative index. Check whether the answer should be larger or smaller than the front number.
The full lesson is waiting for you
Create a free account to unlock activities, practice tests, presentations and videos - and to track your progress and confidence score for every subject.
The full lesson is waiting for you
Create a free account to unlock activities, practice tests, presentations and videos - and to track your progress and confidence score for every subject.