>>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?

>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 mere "convenience". Below, I give my favorite answer to the question "Why is x0 = 1?" I will also try to make my answer appropriate to a middle school audience. Unfortunately, trying to do that will make the answer lengthy, so please bear with me.

>However, most of the time 00 is left undefined.

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.
Therefore, I also briefly address the issue of 0^0 below.

---------------------------------------------------------------

As a preliminary, please consider the simple pattern

20 = 5*4 = 5 + 5 + 5 + 5

15 = 5*3 = 5 + 5 + 5

10 = 5*2 = 5 + 5

5 = 5*1 = 5

What does the next line in the pattern look like? How about

0 = 5*0 =

perhaps? I grant that it looks strange. But my point is that, since
we were removing a term of 5 on the right side each time, there are
now _no_ terms of 5 remaining there. It's an "empty sum", a sum of
no terms. (Of course, you shouldn't really write it as a blank
space. In doing that, I was exercising my pedagogical license!) Why
should an empty sum be 0? Well, 0 is the additive identity (that is,
x + 0 = x for all x). In a sense then, 0 is the additive
"background", what's always there, whether there are any other terms
or not. So after all the term of 5 have been removed, we're left
with just that background, the additive identity. Of course, having
used terms of 5 was not important to the example; any empty sum must
be 0 by the same reasoning, and so we often say "the empty sum",
rather than "an empty sum".

Now let's think about products instead of sums. Consider the pattern

81 = 34 = 3*3*3*3

27 = 33 = 3*3*3

9 = 32 = 3*3

3 = 31 = 3

1 = 30 = 1

I went ahead and wrote out the line involving 30. (You
already knew what it would be. Right?) We were removing a factor of
3 on the right side each time, and so, by the last line, there were
no factors of 3 remaining. We got an empty product, a product of no
factors. Now, since x*1 = x for all x, 1 is the multiplicative
identity, and so 1 is our multiplicative "background", what we can
always think of as being present in a product, whether there are any
other factors or not. Once all the factors of 3 were gone, only the
multiplicative background, 1, remained. But using 3 in our example
was unimportant of course. Any empty product must be the
multiplicative identity, 1, just as any empty sum must be the
additive identity, 0. Thus we normally speak of "the empty
product", rather than "an empty product".

So let's think about x0 in general. How many factors of x
does it have? Well, none, of course; the exponent tells us that it
has 0 factors of x. Since there are _no_ factors of x involved in
the product, our result should be _independent_ of x. And of course
that's exactly what we find from the reasoning in the previous
paragraph: x0 is the empty product, and must therefore be
1, regardless of x.

--- Digression about 0! ---

Have you ever heard of factorials? (I don't know if factorials are
encountered in middle school or not.) For example, 6!, read as "six
factorial", means 1*2*3*4*5*6, and so its value is 720. Generally,
for a positive integer N, we could write N! = 1*2*3*4*...*N. So
let's define N! as being the product of the first N positive
integers. Now at first glance, most people would be inclined to
think that the definition in the previous sentence would be
applicable only when N is a _positive_ integer. But that's not quite
true. Our definition still works when N = 0, and that's good because
it turns out that, in using factorials, we often need to deal with
0! (Before reading further, can you give the value of 0! and explain
why it must have that value?) If we use our definition of factorial,
we find that 0! should be the product of the first 0 positive
integers. Ah! It's the product of no factors, and thus it's the
empty product, and so it must be 1.

I included this digression because students often ask "Why is 0! =
1?" and teachers often reply simply "By definition." But of course
such an answer is ultimately unsatisfying because it doesn't tell
the students _why_ 0! is defined to be 1. My favorite explanation is
that in the paragraph above. So 0! and x0 are seen to be
1 for exactly the same reason: They're both the empty product.

--- End digression ---

Finally, I need to address the matter, raised by Alex, of 00. To deal with this quickly, I could merely repeat what I had said right before the digression: x0 is the empty product, and must therefore be 1, _regardless_ of x. Then, as a special case, we have 00 = 1.

I wish I could just stop with that and say "Case closed." But then you would wonder why Alex had said "However, most of the time 00 is left undefined."

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.

So then you're wondering: Why is 00 often left undefined, at least in lower level texts? Unfortuantely, to explain that gaffe would take us far beyond what's appropriate for a middle school audience. (If anyone's curious, I'd be willing to give my opinion about this matter in either the College Math or This and That part of CTK.)

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..."

Regards,
David W. Cantrell