Here are 16 ways that I came up with to do something with those two digits.
It seems sort of silly to group all of these together. I mean, "32" is an entirely different sort of answer than "2+3."
Let's distinguish between two categories. On the one hand you have the operations, the things that perform some sort of process (or mapping) on two different numbers that yields some third number.
I think that adding, subtracting, multiplying and dividing are all clearly examples of performing some operation on 2 and 3. So all of these go into my "operation" pile.
It really seems to me that "23" and "32" don't belong in this pile. I did some other trick with my starting digits to get 23 and 32. I didn't really perform some process on the digits 2 and 3 to produce a third number. Instead, I used those digits to express a new number. It was just smushing the digits together -- I wasn't combining them with some operation like addition or multiplication.*
* Math Major sez: Oh really? You just performed the mapping (a,b) --> (10a + b). Yeah right, Math Major, yeah right.
This second trick of mine is more about using those starting digits two express a new number. I'm not even treating my starting digits as numbers in their own right, not really. What I'm really doing is treating them like letters of the numerical Alphabet, and lining them up to "spell" some number.
It's obvious to me that "23" and "32" were produced by "spelling" a new number with my starting digits. I think that I also did this with the decimal point.
I've got two categories now: the results of performing some process on numbers, and the results of expressing some number with digits.
This leaves 6 of my 16 results uncategorized.
Next up, I want to tackle the fractions. Some subtlety is needed here, because you can see the fraction bar as a division symbol and it wouldn't be a mathematical sin. It's a bit of a fielder's choice, in that sense. Still, I think of fractions as numbers in their own right, not some sort of quotient waiting to be revealed. I'm going to sort them into "numbers that I expressed" instead of "numbers that I got through performing an operation."
That leaves radicals and exponents. And it's the exponents that I'm really interested in.
If you think that "2 to the third power" means something like "multiply 2 by itself 3 times," then I think that this number pretty obviously goes in the "result of an operation" pile.
The alternative, though, is to think of "2 to the third power" as quite a great deal like "23." It's not so much the result of operating on the numbers 2 and 3, as much as it is using those digits to express some number.
So, which pile does it go in?
Christopher Danielson has pointed out that there is no really good way to talk about the "result" of exponentiation. For addition you get the sum, for division you get the quotient, but how do you talk about what you get out of exponentiation?
I take Danielson's observation as partial evidence that our number language has stacked the deck against thinking of exponentiation as a operation. Instead, the way we talk about exponentiation suggests that it's best seen on the model of "23" or "32," as using digits to express some number.
If this is true, then it partially diagnoses the major problems that students have in learning about exponents. Their teachers have been teaching exponentiation as a process, when it's really a feature of the expressive power of mathematical language.