The three main laws
The laws of indices are shortcuts for multiplying, dividing and taking powers of expressions that share the same base. Instead of writing every factor, you work only with the indices.
Key idea
Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (add the indices). Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the indices). Power of a power: (aᵐ)ⁿ = aᵐⁿ (multiply the indices).
Why adding works
Consider a² × a³. This is (a × a) × (a × a × a) = a × a × a × a × a = a⁵. There are 2 + 3 = 5 factors, which is exactly why we add the indices. Division works the reverse way: a⁵ ÷ a² cancels two factors of a from the top, leaving a³, so we subtract. And a power of a power, such as (a²)³, means a² × a² × a², which gives a⁶ — the same as multiplying 2 × 3.
Worked example
Simplify 2³ × 2⁴ ÷ 2². First multiply: 2³ × 2⁴ = 2³⁺⁴ = 2⁷. Then divide: 2⁷ ÷ 2² = 2⁷⁻² = 2⁵. Finally 2⁵ = 32. So the answer is 2⁵ = 32.
Special cases
Two more rules are useful. Any non-zero base to the power 0 equals 1, so a⁰ = 1. A negative index means a reciprocal, so a⁻² = 1/a². For a product inside a bracket, the power applies to each factor: (x²y³)² = x⁴y⁶.
Remember
- The laws only work when the bases are the same.
- Multiply → add indices; divide → subtract; power of a power → multiply.
- a⁰ = 1 and a⁻ⁿ = 1/aⁿ.