>>>Please can anyone explain to me why, when any real number is
>>>raised to the power of zero, the result is 1.
>>
>>By definition, or you can say by common agreement.
>
>When someone asks "Why?", I've never been very fond of
>answers of that sort, although such answers are indeed
>often given. I'm not fond of them because, to me at least,
>they seem to evade the basic issue:
>_Why_ is it that we commonly agree to define something in a
>certain way?

Well, no one disagrees with you. As you have noticed, the claim about the fact of the definition has been followed by an explanation, however short.

I think it is a good practice to start with 1-2 sentences underlying the jist of the coming response, and then proceed with additional verbiage.

>>It's a convenience actually since then
>>
>>ab·a-b = ab - b = 1.
>
>Yes, that's a reason for why we define it that way. But in
>my opinion, the "real" reason is much more compelling than

You mean an "additional" reason?

>I'm not at all sure that's done most of the time _by
>mathematicians_. But I regret to say that I agree that it is
>done most of the time in elementary (pre-graduate school level)
>texts.

... and any level Calculus book, of course.

>Therefore, I also briefly address the issue of 0^0 below.
>
>we can always think of as being present in a product,
>whether there are any other factors or not.

This is a delicate point. We are not exactly talking of the integers. Would you call x a factor in

x×1/x = 1.

(I understand what you wanted to say and accept that. The above is to underscore a possible ambiguity in your argument.)

>x0 is the empty product, and must
>therefore be 1, regardless of x.

The argument is nice, but by the same token, 0x = 0 regardless of x.

>Since I've already presented an argument presumably showing
>that 00 must be 1, perhaps I should also now, in
>fairness, present reasons for leaving 00
>undefined. But I can't! I am aware of not a single
>compelling mathematical argument for leaving 00
>undefined.

I do not believe that. To me, the ambiguity created by "0x = 0 for all x" is sufficiently compelling to have second thoughts.

>So then you're wondering: Why is 00 often left
>undefined, at least in lower level texts?

With a view to Calculus and for the same reason that 0/0 is left undefined: there are ways and ways to approach 0.

>Unfortuantely, to
>explain that gaffe would take us far beyond what's
>appropriate for a middle school audience.

This is why ... Is it not? I can't be 100% sure, but perhaps the sources chose to avoid giving an explanation that "would take the audience far beyond what's appropriate for a middle school audience."

>I'll close with a quotation from _Concrete Mathematics_ by
>Ronald Graham, Donald Knuth, and Oren Patashnik
>(Addison-Wesley, 2nd ed., 1994) p. 162:
>
>"Some textbooks leave the quantity 00
>undefined... But this is a mistake. We must define
>x0 = 1, for all x..."

With all due respect, I understand their argument and agree that in Discrete Mathematics it's rather a great convenience to have 00 = 0. But to say it's a mistake to have Calculus in mind, is a misteak. (See B. Cipra's or E. J. Barbeau's books.)