What is 3 to the power of one third?

Answer 1

On a calculator enter 3 raised to the 1/3, should look like 3^(1/3), and you get 1.4424957. I don't really know why it comes out as this. (Whatever you do, don't turn 1/3 into a decimal as it will not work.)

Actually it shouldn't matter whether it's a decimal or fraction. (In fact the calculator probably converted 1/3 to decimal form as the first step in the calculation.)

If x is any positive number, and n is a positive integer, then x^(1/n) is the (n)th root of x. So, in this case, the is the 3rd root (i.e. cube root) of 3.

This "x^(1/n)= (n)th root of x" rule may look strange at first. But there are good reasons for it. If m and n are any integers, then x^(mn)=(x^m)^n. So it seems sensible to define ^ for non-integer indices, in such a way that this rule still holds (if possible). If we put m=1/n, we get

x^1=(x^(1/n))^n

But x^1=x, and so this rule tells us that x^(1/n) is an (n)th root of x.