Index Laws: Review and Consolidation

When you multiply powers of the same base, you add the indices. The reason is just counting: 2³ × 2⁴ is three 2s multiplied by four 2s, which is seven 2s altogether, so 2⁷ = 128. Written generally, a^m × aⁿ = a^m+n. Two conditions matter. The bases must match — 2³ × 3⁴ cannot be simplified at all. And the base itself never changes: the answer is 2⁷, not 4⁷. If you ever doubt it, count the factors.

Dividing powers of the same base subtracts the indices, and again the reason is cancelling. 3⁵ ÷ 3² is five 3s over two 3s; two cancel from top and bottom, leaving three 3s, so 3³ = 27. Generally a^m ÷ aⁿ = a^m−n. Order matters here in a way it did not for multiplication: it is always the top index minus the bottom one, so x⁷ ÷ x³ gives x⁴ while x³ ÷ x⁷ gives x⁻⁴. And since a fraction bar is a division sign, the same law applies whenever powers appear in a fraction.

Raising a power to another power multiplies the indices. Look at why: (2³)² means 2³ × 2³, which is six 2s altogether, giving 2⁶ = 64 — and 3 times 2 is 6. So (a^m)ⁿ = a^mn. The part that catches people is when there is a coefficient inside the bracket. In (2x³)², everything inside gets squared, so the 2 becomes 4 and x³ becomes x⁶, giving 4x⁶. Leaving the 2 alone and writing 2x⁶ is the classic mistake.

Zero and negative indices look strange until you follow the pattern downwards. Going down the list 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2, each step divides by 2. Continue and the next must be 1, so 2⁰ = 1. Continue again and you get a half, then a quarter — so 2⁻¹ is one half and 2⁻² is one quarter. That gives two rules: anything to the power zero is 1, and a negative index means take the reciprocal. The vital point is that a negative index does not make the answer negative — 2⁻³ is one eighth, never minus eight.

Five laws, all on one page. Multiplying adds the indices. Dividing subtracts them. A power of a power multiplies them. Anything to the power zero is one. And a negative index means the reciprocal. When a question combines them, deal with brackets first and then combine: (2x³)² × x⁴ becomes 4x⁶ × x⁴, which is 4x¹⁰. Notice the coefficient 4 came from squaring the 2 and then simply stayed — coefficients multiply normally, only the indices follow the laws.