Math pedagogy proposal: algebra students should be taught that there are different operations that are all called exponentiation

in general, i think it's better to draw more connections than to separate things that may be slightly different. This is a very important skill and we should be encouraging it more than we do.

of course there are things you can do with subsets of the domain of any operations. these things are typically taught when they come up. talking about these additional properties is quite important, but don't conflate special case properties with different definitions.
Almost everything there is too advance to cover when you first learn about exponents. Repeated multiplication basically has to be first
because my thread has been buried, i want to update my answer. yes, there are things called exponentiation which have different definitions from others.

however, i think it is wrong to say they are very different things, in the same sense that it's wrong to say multiplication by arbitrary real numbers is very different thing from multiplication by integers. the big difference is not that the definitions are different because in math there are several things called, say, the limit, the expected value, or the Riesz representation theorem with very different definitions/statements. the big difference is that their properties are different, their domains are different, etc. and these affect what is possible with them (like algorithms for computing them).

teaching students to treat every definition as something different and only mentioning that they agree in certain cases doesn't do the concept of generalization enough justice.

a common joke in math is "mathematics is the study of giving the same name to different things and different names to the same thing." but whenever i see two things with the same name, my thought is typically to see why they should be called the same thing and not to go deep into why they're different. maybe that's just me.
There is just one operation called exponentiation, and it is defined by a power series.

e\^x = sum(x\^n / n!) for n from 0 to infinity

All of the cases you describe here are encompassed by the power series definition, including exponentiation of matrices and operators. Of course, the power series definition probably isn't productive to teach in algebra, as you generally don't learn about sequences and series until later.

Regardless, it might be better to go the opposite direction: instead of saying that there are a bunch of different things called "exponentiation", prove that any *new definition* of exponentiation you introduce is *equivalent to* the other definitions you've already been using in a more limited domain. In other words, restricting to, say, the natural numbers, each definition you've provided is a different way of thinking about the *same* operation. And these ways of thinking have different uses, especially in their ability to generalize.

This idea of gradually generalizing an intuitive concept to areas that aren't so intuitive is extremely important in math, and it seems like saying "we create an entirely new operation whenever we do this" misses the point a bit. What's important (and, sometimes, surprising!) is not how these things are different, but how they're *related,* how you get from one to another, and how that process can be used to develop ideas in much broader contexts.
“Exponentiation is the operation such that a\^n·a\^m = a\^(n+m), zero to the zeroth power is one, zero to a negative power is undefined, and any element to the power of one is equal to itself,” covers all of them (including square matrices, if we define “zero” as “the additive identity” and “one” as “the multiplicative unit”).

You would need to introduce ring theory for such an abstract definition to make sense, though. I’m told that, in Europe, Euler’s identity on the complex plane is often taught as the formal definition, and all those other operations (except the one for matrices) as special cases.
> This instantly resolve the often-asked question about why 0^0 is undefined sometimes and equal 1 sometimes, without having to mention calculus before student know calculus. Well, they're just 2 different exponentiation, one with natural number exponent, and one with real number exponent.

0^0 is equal to 1 all the time.
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