We all have been told that x^0 (x to the 0th power) equals 1, but few have looked at why this is true. What does it really mean when we say, take 5 to the 0th power? What does 0 to a power mean--or more to the point, what does 0^0 mean?
If I type x^0 in Wolfram|Alpha, the simple output is 1 (there are some series and integral representations of the answer too, but I have never really cared for that kind of thing (too messy)). But what are we really doing? A power function is short hand for saying take this number and multiply it by itself this many times (i.e. 3^3 = 3 * 3 * 3 = 27). So, when we say 3 to the 1st power, we are simply saying 3 in an abstract way. If you think about multiplication as a series of steps to get to an amount, this makes sense (i.e. 3 * 4 * 5 = 12 * 5 = 60). The easy way to think about powers is as a function. The first input to the function is the number you want to multiply by itself and the second input is the number of times you want to multiply the number by itself: hence 2^4 could be thought of as power(2, 4) = 2 * 2 * 2 * 2 = 16. Therefore, 3^1 is power(3, 1) = 3 not multiplied by anything else, so the value is simply the same as the first input into the function. This is easy enough (I know this is not true for negative powers).
So what does it mean to take something to the 0th power? The simple power function given above breaks down at this point (i.e. power(3, 0) = 3 * ? or something like that). So what does 3^0 equal? We have been told in school that x^0 equals 1, so 3^0 = 1. But why? One way to think about it is that 3^3 = 3 * 3 * 3 = 27, 3^2 = 3 * 3 = 9, and 3^1 = 3. Do you notice a pattern? When we are moving down the power chain, we are simply taking the previous answer and dividing it by the number, hence 3^2 = (3^3)/3. Therefore, we can say:
x^b = (x^a)/x, when a = b + 1 and where x is any number (Real or Complex).
This looks really complex, but it is not. What we are saying is that x to any power is equal to the value of x to a power one above the current one divided by x (i.e. 2^4 = (2^5)/2 or 2 * 2 * 2 * 2 = (2 * 2 * 2 * 2 * 2)/2 (cross off one of the 2s on the right side and you get 2 * 2 * 2 * 2 = 2 * 2 * 2 * 2, which we know is true)). Using this form, 3^0 = (3^1)/3, which is the same as saying 3^0 = 3/3 = 1. Thus, we state in general that:
x^0 = x/x = 1, where x is any number (Real or Complex) other than 0.
Why can we not say that 0^0 = 1? It would make our lives a lot easier. Well, we have also been told that 0 times any number is 0. As such, we can say 0^3 = 0 * 0 * 0 = 0, 0^2 = 0 * 0 = 0, and 0^1 = 0. Following this pattern, we get 0^0 = 0.
So does 0^0 = 0 or 1? Well we have another problem (D'oh). You cannot divide by 0 (I was told in my Number Theory class (MATH 480) that St. Peter keeps track off the number of times you divide by 0. If the number is too high you cannot get into heaven). Why can you not divide by o? We do not know what it means to take a number of things and place them in equal amount of sets that contain 0 of the thing (i.e. if Pirates are splitting up gold, and there is no gold to be split, how much gold does each Pirate get (I assume the answer is that the captain will get split in two)). Since we cannot split a number of things into equal sets containing 0 of the things, then 0^0 = 0/0, which I guess is undefined?!?
The problem is that 0^0 is an Indeterminate, which in the world of Mathematics is marked on the map with the words, "Here be dragons". You can argue that 0^0 = 1, since x/x = 1. You could also argue that 0^0 = 0, since x * 0 = 0. Or you can play it safe and say 0^0 = undefined. I like the third definition because I work in IT and it is safer to catch an error than to just assume an answer. At any rate, I hope you have a better understanding on how the power function works and on what x^0 means.