Just for the record, I thought this passage from [Omar Khayyam's algebra book][1] (p.49) should be here. In particular, it shows how hard it was to to tie  the understanding of powers to geometry

> I say: what algebraists call square-square is an imaginary concept in
> continuous quantities. It has no existence in any way in materialistic
> objects. For continuous quantities, the terms square-square,
> square-cube and cube-cube are used to denote the number (coefficient)
> of the object (variable). Things share the sort of the quantity (all
> variables have the same type of coefficient: a real number), as is
> continuously shown by He who has the ultimate knowledge (God). The
> things that algebraists use to denote objects and quantities are:
> number, root, square and cube. The number has to be taken as an
> abstract concept. It has no existence unless it is individuated by
> things...
> Square-square, which, to the algebraists, is the product of the square
> by itself, has no meaning in continuous objects. This is because how
> can one multiply a square, which is a surface, by itself? Since the
> square is a two-dimensional object (geometrical figure), and
> two-dimensional by two-dimensional is a four dimensional object. But
> solids cannot have more than three dimensions. All objects in algebra
> are generated from these four genera. And anyone who says that algebra
> is a trick to determine unknown numbers is wrong. So don’t pay
> attention to these people. It is true that algebra and equations are
> geometrical things...


  [1]: https://archive.org/details/AnEssayByTheUniquelyWiseabelFathOmarBinAl-khayyamOnAlgebraAnd