We know that anything raised to the first power equals itself.
For example, (x)^1 = x. We also know that anything raised to the zero power equals 1. For example, (x)^0 = 1. Why is that true for both cases?
Well, makes sense, does it not? If it also follows that . One x multiplied together (so to speak) is just x.
Similar logic applies to . Zero x's multiplied together is obviously 0. [EDIT: Wow, I've handled x^0 enough times you'd think I wouldn't mess this up! x^0 = 1 as explained below. Let that be a lesson to me!]
Do note that 0^0 is a little tougher to settle. Why? An explanation can be found here:
sci.math FAQ: What is 0^0?
We can get this result by forming the new index from 5-3.
Basically 3 of the x's in the denominator "cancel" 3 of the x's in the numerator to give 1.
Hence we can write the "general" way as shown in earlier posts.
Therefore, in index form... if we subtract the indices.
is 1 because it is an "index" way to write a number divided by itself.
Following on from this, you will see why negative indices represent a power of a number divided by a higher power of the same number.