Fractional Indices

So far the index has always been a whole number. But an index can also be a fraction — and a fractional index means a root. A power of \tfrac{1}{2} is the square root, and more generally a power of \tfrac{1}{n} is the nth root:

a^{1/2} = \sqrt{a}, \qquad a^{1/n} = \sqrt[n]{a}

Why does a^{1/2} mean the square root? Because the indices add when you multiply powers of the same base, so a^{1/2} \times a^{1/2} = a^{1/2 + 1/2} = a^1 = a. The number that gives a when multiplied by itself is exactly \sqrt{a} — so a^{1/2} and \sqrt{a} are the same thing.

What about a general fraction \tfrac{m}{n}? It says “take the nth root, then raise to the mth power”:

a^{m/n} = \left(\sqrt[n]{a}\right)^{m}

For example, 27^{2/3} = \left(\sqrt[3]{27}\right)^{2} = 3^2 = 9. Taking the root first keeps the numbers small and friendly — much easier than working out 27^2 = 729 and then taking its cube root.

A fractional index is a root:

A power of a half is the square root — a^{1/2} = \sqrt{a}.
A power of \tfrac{1}{n} is the nth root — a^{1/n} = \sqrt[n]{a}.
A power of \tfrac{m}{n} is the nth root raised to the mth power — a^{m/n} = \left(\sqrt[n]{a}\right)^{m}.
Take the root first to keep the numbers small.