Saved in the math homework!! Brian (yes, that's right, 4th grade) had a paper about all of the ways to write multiples of ten, including exponents. He had to look at all the ways to write them, and put them in the appropriate box. For example, 10,000 had 10*10*10*10, 10 1,000's, and 10 to the fourth power. But there was one, as simple as the number 1, that I didn't know. 10 to the first power, and 10 to the zero power. I thought I knew the first was ten, but how did either one arrive at the number one. All the other choices were ruled out, I just couldn't be sure. So I googled it.
Welcome into my life Purplemath.com. There is a rule about exponents of zero, anything to the power of zero equals just 1. So I say this to Brian, and he writes his answer and moves on. And then I "turn" the page, to find out what the heck they are talking about, and sure enough, it is explained as follows:
Anything to the power zero is just "1".
Why is this so? There are various explanations. One might be stated as "because that's how the rules work out." Another would be to trace through a progression like the following:
(Remember, the second digits are exponents, so it's 3 to the fifth = 3 to the sixth divided by...)
35 = 36 ÷ 3 = 243
34 = 35 ÷ 3 = 81
33 = 34 ÷ 3 = 27
32 = 33 ÷ 3 = 9
31 = 32 ÷ 3 = 3
Then logically 30 = 31 ÷ 3 = 3 ÷ 3 = 1.
A negative-exponents explanation might be as follows:
m0 = m(n – n) = mn × m–n = mn ÷ mn = 1
...since anything divided by itself is just "1".
Another comment: Please don't ask me to "define" 00. There are at least two ways of looking at this quantity:
Anything to the zero power is "1", so 00 = 1.
Zero to any power is zero, so 00 = 0.
As far as I know, the "math gods" have not yet settled on a "definition" of 00. In fact, in calculus, "00" will be called an "indeterminant form". If this quantity comes up on class, don't assume: ask your instructor what you should do with it.
My favorite words in math: "because that's how the rules work out."
That's what I love about math. It just is. The long drawn out explanations can be found out, but really, when it gets down to brass tacks, it just is.