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)... 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.  

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