In this series of posts, we have seen that the definition of and saw that the definition was a little different depending if is even or odd:
- If is even, then means that and . In particular, this is impossible (for real ) if .
- If is odd, then means that . There is no need to give a caveat on the possible values of .
Let’s now consider the definition of , where and are positive integers greater than 1. Ideally, we’d like to simply defined
This definition reduces to previous work (like a good MIT freshman), using prior definition for raising to powers that are either integers or reciprocals of integers. Indeed, if , there is absolutely no ambiguity about this definition.
Unfortunately, if , then a little more care is required. There are four possible cases.
Case 1. and are odd. In this case, there is no ambiguity if is negative. For example,
Case 2: is even but is odd. Again, there is no ambiguity if is negative. For example,
Case 3: is odd but is even. In this case, is undefined if . For example, we would like to be equal to , but is undefined (using real numbers).
Case 4. and $latex $n$ are both even. This is perhaps the most interesting case. For example, how should we evaluate ?. There are two legitimate choices… which lead to different answers!
Option #1: If we just apply the proposed definition of , we find that
Option #2: We could first reduce to lowest terms:
So… which is it?!?!?!?! The rule that mathematicians have chosen is that simplifying the exponent takes precedence over the above definition. In other words, the definition should only be applied in has been reduced to lowest terms in order to remove the above ambiguity.
For the sake of completeness, I note that the above discussion restricts our attention to real numbers. If complex numbers are permitted, then things become a lot more interesting. If we repeat a few of the above calculations using complex numbers, we get answers that are different!
The explanation for this surprising result is not brief, but I discussed it in a previous series of posts: