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)... 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...
Edit. René's post and Joël's comment gave me some new insight about Khayyam's understanding of powers higher than three. Of course, he was aware of them as he explains how a certain equation of power 4 can be solved:
Now, whoever said: square-square plus three squares equals twenty-eight; he halved the squares then multiplied it by itself and then added the number; and took the root of the result to equal five and a half; then subtracted half the squares to get four which the square, and the square of the square is sixteen...
But, for him algebra and equations were attached to geometry. Apart from number that "has to be taken as an abstract concept", $x$ , $x^2$, and $x^3$ had geometrical meaning, side, square , and cube, respectively. Thus, immediately after mentioning the solution of the equation above, he warns the reader as follows:
...and he thought that he deduced the square of the square using algebra: is very feeble in his thinking. This is because he did not deduce the square of the square but rather he deduced the square.It is exactly as if he said: square plus three roots equals twenty-eight, then he determined the root using the second reduction, and concluded that the square of this root is the square of the square, which is a secret from which you will come to know other secrets.
All in all, it is a good example of how a "philosophical" belief could impede the advance of knowledge even for such an intelligent mind.