Thank you very much for those pointers - and for the encouragement (yet!). I will keep your advice in mind. I think my big problem is the meeting in the middle - you know where to start, and where you want to finish, it's the meeting in the middle that I can't find!
The rational exponents explanation is delightful. That was fun. I first sat and carefully thought through: Why that condition on that variable? Ok, zero raised to any power is zero, so wouldn't it still work for zero? But then I thought, except for raising to the power zero, 00 = 1... so does it still work?
02/3 = [MATH]\sqrt[3]{0^2}[/MATH] = 0
but
00/3 = [MATH]\sqrt[3]{0^0}[/MATH] = [MATH]\sqrt[3]1[/MATH] = 1 and if we cube both sides
(00/3)3 = 01=0
but
([MATH]\sqrt[3]{0^0}[/MATH])3 = 00 = 1.
So that's why zero was excluded?
JeffM's constraints work, logically they are correct, but they don't show the full range of numbers for which the identity holds. It works under other circumstances too.
See this post about "power of power" and contrainsts (click)
I'm guessing that it would become increasingly difficult to mathematically
prove that this property holds under all of these different circumstances.
It is tempting to adopt, and many mathematicians feel no difficulty in adopting, a Platonist view of mathematics. (Platonism taken to its ultimate conclusion is the philosophy that your mother has never really existed but is merely the shadow of the one real mother that no living human has ever known.) In that view, the human mind
discovers mathematics. The other view, the one that I prefer, is that the human mind
creates mathematics, in part by assuming that observable regularities in the physical world are invariably true in the idealized world of the intellect. The Platonists assume that the invisible realm is logically consistent and thus discoverable by reason. The non-Platonist requires that what we create be logically consistent. In other words, both agree that valid mathematics is logically consistent, and thus the two schools can live in reasonable amicability. For non-Platonists like me, definitions are free exercises of the mind and judged solely on utility and consistency with other definitions.
Notice that I did not define [MATH]r^a.[/MATH] Obviously my definition of a rational exponent depends on my definition of an integer exponent so what I said before was incomplete. Here is a set of definitions that I like for integer exponents
[MATH]\text {Given } r \in \mathbb R, \ a \in \mathbb Z, \text { and } r > 0, \text { then}[/MATH]
[MATH]a = 0 \implies r^a \equiv 1, \text { and } r^a \equiv r * r^{(a - 1)}.[/MATH]
Now why did I pick that definition?
Mainly because it has the very nice
consequence that any integer power of r is a real number. If I allowed r to be 0, then negative powers would not be real numbers. Moreover if I allowed r to be less than 0, then some rational exponents would not be real numbers.
Thus, the reason that I stipulated that r > 0 for rational exponents is that I had implicitly stipulated that for integer exponents.
In fact, if you are willing to restrict exponents to non-negatice rational numbers, there is no reason to restrict r to positive real numbers; it will work just fine to restrict r to non-negative real numbers. In particular, if r is a real number, q is a rational number, and both are non-negative, then no problem arises from a definition of r^q that entails that 0^0 = 1. However, mathematicians may want to work with real functions to a power that is a real function. In that case, allowing both functions to be zero simultaneously can cause problems in the field of mathematics known as analysis. Thus, some mathematicians define things so 0^0 is not defined, and some mathematicians define things so that 0^0 = 1. For more details, see
en.m.wikipedia.org
As cubist correctly said, we can define exponents so that they make sense and are useful with fewer restrictions than I gave. If we do so, however, the laws of exponents become more complex. If we define exponents in a way that restricts r to the positive reals, we get a set of laws for exponents with no exceptions. In practice, I use exponents with numbers that are not positive reals only after checking that I am not about to be bitten by the exceptions.