Index Rules

An index (also called exponent or power) is a repetition counter for multiplication: 3⁵ means 'multiply five copies of 3' = 243 — emphatically not 3 × 5 = 15. The big number is the BASE, the little one the INDEX. It works identically for letters: a⁴ = a × a × a × a. Hold onto the copy-counting picture: every index law in this lesson is just counting copies cleverly instead of writing them all out.

Law 1: multiplying powers of the SAME base — count the copies. a³ × a⁴ is (three copies) times (four copies) = seven copies = a⁷. So a^m × aⁿ = a^m+n: multiply the powers, ADD the indices. With coefficients, the jobs split: 2x³ × 5x⁴ = 10x⁷ — numbers multiply, indices add. It works for number bases too: 2³ × 2⁴ = 2⁷ = 128 (check: 8 × 16 = 128 ✓).

Law 2: dividing powers of the same base — the bottom copies cancel top copies. a⁵ ÷ a² leaves 5 − 2 = 3 survivors: a³. So a^m ÷ aⁿ = a^m−n: SUBTRACT the indices. With coefficients: 12x⁶ ÷ 3x² = 4x⁴ — numbers divide, indices subtract. Last lesson you did this by writing copies out and cancelling; now the law does the counting for you.

Law 3: a power OF a power. (a³)⁴ means four copies of a³, and each of those is three copies of a — twelve copies in all: (a³)⁴ = a¹². So (a^m)ⁿ = a^m×n: MULTIPLY the indices. Crucial extra: the outside power hits EVERYTHING inside the bracket, coefficient included. (2x³)² = 2² × (x³)² = 4x⁶ — the 2 gets squared too. Notice the pattern across the laws: × powers → add, ÷ powers → subtract, power of power → multiply.

The zero power looks weird until you corner it: compute a³ ÷ a³ two ways. Law 2 says it's a³⁻³ = a⁰; common sense says anything divided by itself is 1. Both must be right, so a⁰ = 1 — always, for any non-zero base. 7⁰ = 1, x⁰ = 1, (3ab)⁰ = 1. The precision point: the index only grabs what it touches, so 5x⁰ = 5 × x⁰ = 5 × 1 = 5, while (5x)⁰ = 1. Brackets decide the victim.

One more corner to explore: what if subtracting indices goes BELOW zero? Compute a² ÷ a⁵ two ways. Law 2 says a²⁻⁵ = a⁻³; writing the copies out, two a's cancel against five, leaving three underneath: 1/a³. Both must agree, so a⁻ⁿ = 1/aⁿ — a negative index means RECIPROCAL, 'one over'. Examples: 2⁻³ = 1/2³ = 1/8, and 5x⁻² = 5/x² (the index only grabs the x, so the 5 stays on top). The laws you already know handle negative indices unchanged: x³ × x⁻⁵ = x⁻² = 1/x².

The laws on one card: multiply powers → ADD indices; divide → SUBTRACT; power of a power → MULTIPLY; zero power → 1; and negative index → reciprocal (a⁻ⁿ = 1/aⁿ). They only ever apply to the SAME base. Combos chain them one move at a time: (2a²)³ × a ÷ a⁴ — first the bracket: 8a⁶; then × a: 8a⁷; then ÷ a⁴: 8a³. When in doubt, retreat to copy-counting: write the copies out and the law re-derives itself. Quiz and exercises!