Punch 9^{0.5} into a calculator and it answers 3. Try 8^{1/3} and out comes 2. That looks impossible — how can you multiply a number by itself "half a time"? You can't, at least not in the counting-on-your-fingers way. Yet the calculator is completely right, and the reason is one of the neatest tricks in all of arithmetic.

Here is the punchline first: a fractional index is a root. A power of \tfrac{1}{2} means the square root, a power of \tfrac{1}{3} means the cube root, and so on. So 9^{1/2} = \sqrt{9} = 3 and 8^{1/3} = \sqrt[3]{8} = 2. Roots and powers turn out to be the same idea wearing two different costumes.

Why a half must mean "square root"

Nobody just decided this to be cute. It is forced on us by the laws of indices you already know — the rule that when you multiply powers of the same base, the indices add.

Suppose a^{1/2} is going to mean something. Whatever it is, if we multiply it by itself the indices must add:

So a^{1/2} is a number that gives a when multiplied by itself. But that is the exact definition of the square root! There is no wiggle room — if the index laws are to keep working, a^{1/2} is obliged to be \sqrt{a}.

The same argument works for any root. Multiply a^{1/3} by itself three times and the indices add to 1:

so a^{1/3} is the number you cube to get a — the cube root. In general:

Worked examples: the simple roots

Pick perfect squares and cubes and these come out clean. Read the denominator of the index as "which root", then just recall your times tables:

• 25^{1/2} = \sqrt{25} = 5, because 5\times5=25.
• 64^{1/2} = \sqrt{64} = 8, because 8\times8=64.
• 27^{1/3} = \sqrt[3]{27} = 3, because 3\times3\times3=27.
• 1000^{1/3} = \sqrt[3]{1000} = 10, because 10^3=1000.

The full fraction: a^{m/n}

What if the top of the fraction isn't 1? A general index \tfrac{m}{n} says: take the nth root, then raise to the mth power. The denominator is the root, the numerator is the power:

Why is this allowed? Because \tfrac{m}{n} = m \times \tfrac{1}{n}, and the power-of-a-power law says you multiply the indices: a^{m/n} = \left(a^{1/n}\right)^{m} = \left(\sqrt[n]{a}\right)^{m}.

Example 1 — 8^{2/3}. The denominator 3 says cube root; the numerator 2 says square it:

Example 2 — 16^{3/2}. The 2 says square root; the 3 says cube it:

Example 3 — 32^{2/5}. The 5 says fifth root (2^5=32); the 2 says square it:

Negative and fractional together

You can stack the ideas. A negative index means "reciprocal" (flip it over) and a fractional index means "root" — so a negative fraction means flip it and take the root:

Notice how each rule you meet just slots into place beside the others — nothing is thrown away.

When the answer isn't whole

Perfect squares and cubes give tidy whole answers, but most bases don't. What is 2^{1/2}? It is \sqrt{2}, and \sqrt{2}=1.41421\ldots — a never-ending, never-repeating decimal. That is completely fine. A fractional index is still a perfectly good number; it just happens to be irrational, so we round it when we need a figure:

These "in-between" values are the whole point of fractional indices: they fill the gaps between the whole-number powers so densely that the powers of a base join up into one smooth, unbroken curve.

Checking your work — run it backwards

Every root claim can be checked by raising it back to the power. If 8^{2/3}=4 is right, then multiplying the index by \tfrac{3}{2} should undo it and return the 8-side. More simply, verify the root itself: you claimed \sqrt[3]{8}=2, so check 2^3=8 — yes. Then 2^2=4. Done, with confidence.

It is also worth knowing that the root and the power can be done in either order — the value is the same:

For 8^{2/3}: root-first is (\sqrt[3]{8})^2 = 2^2 = 4, and power-first is \sqrt[3]{8^2} = \sqrt[3]{64} = 4. Same answer — but root-first kept the numbers small, which is why we recommend it.