The idea

Fractional indices

Why a power of 1/n is an nth root, how m/n means "root, then power", how negative fractional indices chain reciprocal + root + power, and how to solve equations by matching bases.

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Number · Indices

Fractional indices

Why a power of 1/n is an nth root, how m/n means "root, then power", how negative fractional indices chain reciprocal + root + power, and how to solve equations by matching bases.

Why it works

Why a^(1/2) is the square root

Why $a^{1/2}$ is the square root — not half of $a$ . The multiplication law forces it:

9^{1/2} \times 9^{1/2} = 9^{\frac{1}{2} + \frac{1}{2}} = 9^1 = 9.

9^{1/2}

is the number that multiplies by itself to give 9 — that is exactly what

\sqrt{9} = 3

means. It is not

4.5

: the index halves the power count, not the value. Likewise

a^{1/3} = \sqrt[3]{a}

(three copies multiply to

a

), and in general

a^{1/n} = \sqrt[n]{a}

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