This question already exists, but its answer was not helpful to me.
If the elements of a set are of the form $((a,b),(c,d))$ (the elements are pairs of pairs), and the like, how does one denote that? A set of $n$-tuples consisting of numbers all ranging across the exact same set $S$ can be denoted as $S^n$. But what to do if it consists of $x$-tuples of $y$-tuples of $z$-tuples of... etc.?
Strictly speaking, $S^{ab}$ and $(S^a)^b$ are not the same. There is, however, a canonical bijection between them given by $$(x_1,...,x_{ab})\mapsto ((x_1,...,x_a), ..., (x_{ab-a+1}, x_{ab}))$$ which we often "mod out" by implicitly (note that this is really the same issue as the technically-non-associative nature of the Cartesian product). In situations where the difference is something you actually want to keep track of as opposed to an annoyance you want to avoid, you obviously don't want to do this. In terms of communicating clearly to the reader, all you need to do is make an explicit remark somewhere pointing this out, so that your readers understand why you aren't "simplifying" the relevant expressions.