I've been noticing how many web pages there are about high school algebra that give examples of taking fractional powers of negative numbers. I think the standard axiomatic approach to the real numbers is only to define fractional exponents for positive numbers. Many of the pages do this, but they then proceed to give examples of taking fractional powers of negative numbers. These pages also state the property of exponents that says (x^a)^b = x^(ab). But this is false if (-1)^(2/3) = 1. ( Let x = -1, a = 2/3, b= 3/2).
Equally amusing is to try discussing this topic with people trained in higher mathematics. They began talking about complex numbers, the solutions to x^n = 1, etc. However, the question here is how to define an operation in the axiomatics of the real numbers. An operation may either be undefined or it may produce a unique result. This isn't a question about the set of roots for an equation.