When I was a kid, my dad asked me to explain fractional exponents, and perhaps any non-positive integer exponents, to him. He objected to the idea of multiplying something by itself 1/2 times.
I failed to answer the question to his satisfaction. My own son is now reviewing the rules of exponentiation, and it occurred to me (30 years later) why my explanation to Dad failed.
Essentially, there's a small bait and switch required, and my dad didn't fall for it.
The meaning that my dad gave to exponentiation was that x^n equals x times itself n times.
Using this rule, it is easy to demonstrate that x^a x^b = x^(a + b), and this can be used to justify expressions like x^(1/2). However, doing this really means that we've switched the definition of exponential, defining an exponential as any number that satisfies the relationship:
x^a x^b = x^(a+b)
where x^1 = x. This slight of hand is required to give meaning to x^(1/2) or other exponentials where the exponential argument is any non-positive integer.
