Laws of Positive Integral Indices

1Summary 重點摘要

Learn a few index rules so you don't have to calculate powers the long way.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What an index and a base areAn index is a compact way of writing repeated multiplication: 2³ = 2 × 2 × 2 = 8, where 2 is the base and the small 3 on the top right is the index. The position matters, because writing the 3 beside the 2 instead of above it changes the meaning entirely. A few basic results are worth memorising: any number to the power 1 is itself, 1 to any power is still 1, and 10 to the power n is a 1 followed by n zeros.
2. Step 2: Same base multiply — add the indicesThe first law states aᵐ × aⁿ = aᵐ⁺ⁿ: when multiplying powers with the same base, add the indices. The idea is simple — 2³ × 2² means three 2s multiplied by two more 2s, which is five 2s in total, giving 2⁵ = 32. The base never changes, so 2³ × 2² equals 2⁵ and not 4⁵. If coefficients appear, handle them separately and then combine, for example 3a² × 2a³ = 6a⁵.
3. Step 3: Same base divide — subtract the indicesThe second law states aᵐ ÷ aⁿ = aᵐ⁻ⁿ: when dividing powers with the same base, subtract the indices. In 2⁵ ÷ 2², two of the five 2s cancel, leaving 2³ = 8. The order of subtraction cannot be reversed: the divisor's index is taken away from the dividend's, since 2² ÷ 2⁵ = 12³ = 18 is completely different from 2⁵ ÷ 2² = 2³ = 8. Like the first law, this only applies when the bases are identical.
4. Step 4: A power of a power — multiply the indicesThe third law states (aᵐ)ⁿ = aᵐˣⁿ: when a whole power is raised to another power, multiply the indices. Expanding shows why: (2³)² = 2³ × 2³, and by the first law this is 2³⁺³ = 2⁶ = 64, exactly 3 × 2. The most common error is adding the indices to get 2⁵, so keep in mind that repeated multiplication adds indices while a power of a power multiplies them. If the bracket contains a coefficient, raise it too: (2a²)³ = 8a⁶.
5. Step 5: Combining the laws to simplifyFor a more complicated expression, combine the three laws and simplify step by step, usually dealing with multiplication before division. Take (2³ × 2⁴) ÷ 2²: add the indices inside the bracket to get 2⁷, then subtract to get 2⁵ = 32, with no need to expand anything. Write one law per line and show clearly how each index changes; with nested brackets, work from the inside out and apply only one law at a time. The worked example (3² × 3⁴) ÷ 3³ simplifies the same way to 3³ = 27.
6. Step 6: The same base is the preconditionAll three laws share one precondition: the bases must be exactly the same. With different bases such as 2³ × 3², the indices must never be combined; evaluate each power separately to get 8 × 9 = 72. Checking the base should be the first move in every question — confirm the bases match before applying any law. A common examination trap is confusing 2³ × 2² with 2³ × 3², so circle every base before you start; this simple habit prevents such lost marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Same base multiply, add indices
• Same base divide, subtract indices
• Power of power, multiply indices
• Only works when base same
• Using formulas simplifies much faster