Does anyone know why any number to the power of 0 equals 1? I had it figured out a long time ago, and it’s so that another operation could work, but I can’t remember!!

There might be other reasons which aren’t coming to me at the moment since it’s 8:00 on a Saturday morning, but the first thing that came to me was powers of ten.

10[sup]4[/sup] = a 1 followed by 4 zeroes

10[sup]3[/sup] = a 1 followed by 3 zeroes

10[sup]2[/sup] = a 1 followed by 2 zeroes

10[sup]1[/sup] = a 1 followed by 1 zeroes

Therefore,

10[sup]0[/sup] = a 1 followed by no zeroes

Somebody else (possibly me) will be along shortly to present a more detailed reason.

Caffeine - taking - effect… eyes openeing… muscles flexing… I am- **AWAKE!** *(cape flapping in breeze)*

Now then. Flipping through some of my moldy oldies proof books for mathematics, this is all I could come up with:

In order to be able to generalize the algebraic rule that: a[sup](b + c)[/sup] = a[sup]b[/sup] × a[sup]c[/sup], for example, even when b = 0, we have to set a[sup]0[/sup] = 1. Therefore, it is the convention that a[sup]0[/sup] = 1. This convention is needed in order to have consistency in this algebraic rule of operations.

An easy way to think of it is this:

3[sup]3[/sup] = 27

3[sup]2[/sup] = 9 **or** 3[sup]2[/sup] = 3[sup]3[/sup]/3

This of course, holds for all numbers, giving the following generalization:

a[sup]x[/sup] = a[sup]x+1[/sup]/a

*exception - when a = 0, weird stuff happens

So, 3[sup]0[/sup] = 3[sup]1[/sup]/3, or 3/3, or 1.

I think this qualifies as a proof, or at least a general property of powers, since by this, 3[sup]-1[/sup] = 3[sup]0[/sup]/3, or 1/3, which is quite correct.

Here’s a threadfrom last year in which this point is discussed:

http://boards.straightdope.com/sdmb/showthread.php?threadid=28327

This should answer your questions.

What about negative and fractional exponents? I get negative exponents, but how are fractional exponents calculated?

of course, there is no such thing as multiplying 3 by itself -1 or 1.5 times, is there?

Wait, forgot. Fractional exponents have to do with square roots. Gee, it’s been so long since I’ve taken a math class. <takes out precalc book>

I’ve always thought this was an intuitive way of thinking of it:

What do you get if you add three to itself zero times? I would argue that it’s zero, since zero is the additive identity (the “nothing” of addition). So an “empty sum” is zero.

What do you get if multiply three by itself zero times? Again, I would argue that it’s one, since one is the multiplicative identity (the “nothing” of multiplication). So an “empty product” is one.

If you think of X[sup]n[/sup] as the product of n copies of X multiplied together, then X[sup]0[/sup] makes no sense. The definition of exponentation must be extended to allow exponents that are not positive integers. We want to extend the definition so that X[sup]n[/sup] still follows as many of the same rules as the product defintion. One rule is:

X[sup]m[/sup]/X[sup]n[/sup] = X[sup]m-n[/sup], for m>n

If we allow m = n we get,

X[sup]m[/sup]/X[sup]m[/sup] = X[sup]m-m[/sup]

1 = X[sup]0[/sup]

X[sup]0[/sup] must equal one so that the rules still apply.