Just for the record, I thought this passage from Omar Khayyam's algebra book (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...