Multiplying and dividing
- When multiplying powers of the same number, add the indices: a³ × a⁴ = a⁷.
- When dividing, subtract the indices: a⁸ ÷ a² = a⁶.
- Multiply any numbers in front separately: 4x² × 3x⁵ = 12x⁷.
Powers of powers
- When raising a power to another power, multiply the indices: (a³)² = a⁶.
- (2x³)² = 4x⁶: square the 2 as well.
Zero and negative indices
- Anything to the power 0 is 1: a⁰ = 1, 5⁰ = 1.
- A negative index means 'one over': a⁻ⁿ = 1 ÷ aⁿ. So 2⁻³ = 1/8.
Fractional indices (Higher)
- A fractional index means a root: a^(1/2) = √a and a^(1/3) = ∛a. So 25^(1/2) = 5.
- a^(m/n) means take the nth root, then raise to the power m: 8^(2/3) = (∛8)² = 2² = 4.
- Negative fractional indices: 16^(−1/2) = 1 ÷ √16 = 1/4.
Key terms
- Index
- The power a number is raised to (plural: indices).
- Base
- The number being raised to a power.
- Index laws
- Rules for multiplying, dividing and raising powers of the same base.
- Negative index
- A power that means the reciprocal: a⁻ⁿ = 1/aⁿ.
- Fractional index
- A power that means a root: a^(1/n) is the nth root of a.
- Reciprocal
- One divided by a number.