What is a more rigorous definition for powers of rational numbers?

Question

My current understanding of $x^\frac{m}{n}$ is that it is equal to $\sqrt[n]{x^m}$. Now technically, (-1)$^\frac{2}{4}$ is equal to $\sqrt[4]{(-1)^2}$=1. As $\frac{2}{4}$=$\frac{1}{2}$, (-1)$^\frac{1}{2}$ is technically equal to 1. However, the square root of -1 is i, not 1. How do I resolve this argument? I'm guessing the rule for raising numbers to rational powers applies only to irreducible fractions but is that it, or is there more to it?

Answer 1

Defining $x^{m/n}$ as $\sqrt[n]{x^m}$ is only done when $x\geqslant0$, not for every $x\in\Bbb R$. Otherwise, we will run into trouble, as you have noticed. Actually, we can still define $x^{m/n}$ as $\sqrt[n]{x^m}$ when $x<0$, but only when $n$ is odd.