Example: Investigate for which numbers x does the power x^x make sense.

Solution: Recalling what we learned in the section powers as numbers in Theory - Elementary function, we see right away that such power makes sense for x > 0. What happens if we look at other values of x? The first that comes to mind is x = 0. Recall that we defined the zero'th power as A⁰ = 1 for any real A, so in particular 0⁰ = 1 and x = 0 can be used in the investigated power.

It gets worse when we look at negative x. For some x we have no trouble, for instance negative integers are fine, taking n = −m we get nⁿ = (−1)^m/m^m. The next largest class is the negative rational numbers, how do we stand there? Quite badly. Consider x = −1/m for some positive integer m. If m is odd, we have no problem, we get x^x = −[m^1/m] and the m-th root of a positive number is well-defined. On the other hand, if m is even, the the power x^x makes no sense since we cannot make an even root of a negative number.

It follows that all negative rational numbers with even denominators are out, and since such numbers form a dense set on the negative half-line, it follows that we cannot form a nice set were x^x would work. It also means another thing. For B irrational we defined A^B by approximating B using rational numbers. Since near every negative irrational number we have lots of rationals with even denominators, the approximating procedure fails. Consequently the number x^x makes no sense for all negative irrationals as well.

Summary: We found out that the expression x^x makes sense as a number for x = 0, for all positive x and for those negative x that are rational numbers with odd denominators.

Note: If we tried to find the domain of x^x as a function by answering the question "when does it make sense", we would get the set described above. The negative part of such set is extremely ugly from the point of view of analysis. There are no continuous intervals, just lots of individual points, so when working with such a function, we would not be able to find limits or derivatives on the negative part. This shows that this approach to general powers does not yield reasonable answers. Note though that the positive part of the "domain" we just found is fine. Not surprisingly, when we determine the domain of x^x properly, we get exactly the positive reals.