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Number · Indices

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The index laws

Why multiplying powers adds the indices, dividing subtracts them, and a power of a power multiplies them — and why every part of a bracket, coefficient included, takes the power.

Number · Indices

The index laws

Why multiplying powers adds the indices, dividing subtracts them, and a power of a power multiplies them — and why every part of a bracket, coefficient included, takes the power.

Why it works

A power is just repeated multiplication: 232^3 means 2×2×22 \times 2 \times 2. Every index law falls out of counting the factors — none of them needs memorising once you can see the count.

Multiplying powers adds the indices.

23×24=(2×2×2)×(2×2×2×2)=23+4=27.2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^{3+4} = 2^7.

Three 2s times four 2s is seven 2s — the indices add. They do not multiply (2122^{12} is wrong), and the base does not change (474^7 is wrong): you still only ever multiply 2s.

Dividing powers subtracts the indices.

x9x3=x⋅x⋅x⋅x⋅x⋅x⋅x⋅x⋅xx⋅x⋅x=x9−3=x6\frac{x^9}{x^3} = \frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x} = x^{9-3} = x^6

— three of the xxs cancel top and bottom, so three get knocked off the count.

A power of a power multiplies the indices.

(a3)4=a3×a3×a3×a3=a3×4=a12.(a^3)^4 = a^3 \times a^3 \times a^3 \times a^3 = a^{3 \times 4} = a^{12}.

Four copies of "three aas" is twelve aas. The classic error is adding (a7a^7) — but adding is what multiplication of powers does; raising a power repeats the whole block.

The laws need the SAME base. 23×522^3 \times 5^2 is not 10510^5 or anything tidy — there is no shared factor to count. But numbers often hide a common base: 8=238 = 2^3 and 4=224 = 2^2, so 82×43=(23)2×(22)3=26×26=2128^2 \times 4^3 = (2^3)^2 \times (2^2)^3 = 2^6 \times 2^6 = 2^{12}. Rewriting everything as a power of the smallest base is the standard exam move.

Everything in a bracket takes the power — the coefficient too.

(3x2)3=3x2×3x2×3x2=27x6.(3x^2)^3 = 3x^2 \times 3x^2 \times 3x^2 = 27x^6.

Three 3s multiply to 2727, and the squares add up to x6x^6. Leaving the coefficient alone (3x63x^6) or squaring it out of habit (9x69x^6) are the two marks-losers here: the power outside the bracket applies to every factor inside.