What makes multiplication a basic operation on natural numbers but exponentiation is not?

Question

Dear StackExchange Math Community: It has puzzled me for some time why multiplication is considered a basic arithmetic operation on natural numbers, but exponentiation is just viewed as a shorthand of repeated multiplication? My understanding is multiplication, like exponentiation, is simply a shorthand for repeated addition and not a distinct operation on natural numbers. Please explain. Thank you in advance.

Answer 1

An arithmetic operation is a function from one or more (some say zero or more) elements of a set $A$ to an element of the same set $A.$ The encyclopedia that inspired your question says this. Whether one calls an arithmetic operation "basic" is (as far as I'm aware) simply a matter of opinion with some vague constraints, much like deciding whether a mathematics problem is "easy" or "difficult".

Whether or not you can use operation "$+$" to define "$\times$" is irrelevant to whether "$\times$" is an operation. Technically, it is sufficient that "$\times$" takes two numbers as input and produces a number as output. But in practice, what makes some functions "operations" and other functions just "functions" is a matter of habit, convention, and convenience.

To be clear, however, exponent is just what we call the symbol $b$ in the expression $a^b,$ not the whole expression, as described in the Comment on the encyclopedia's page on "Exponent". That is, an exponent is just an expression that occurs within the operation called exponentiation, the same way divisor is an expression that occurs within the operation called division.

Why it says earlier on that page of the encyclopedia that exponent is the same as exponential function is a mystery to me; I have never seen anyone use the word "exponent" in that way.

Referring to the encyclopedia again, the page on the Ackermann function mentions "addition, multiplication, exponentiation, and all higher-order analogues of these operations", implying that exponentiation is considered an operation much like addition or multiplication.

For some reason, however, there is apparently no page in the encyclopedia named "Exponentiation". I don't know why not, but you should keep in mind that no encyclopedia is really complete (or, for that matter, authoritative: if you need to establish a fact for a serious research paper, look elsewhere for a citation).