Ok, imagine having a finite line segment from point (a) to point (b) in $R^2$ . I'm not familiar with mathematical terminology of this kind, but let me state that the line we began with is the geometric interpretation of $A^1$. The geometric interpretation of $A^2$ is a square with sides $A^1$. We could go on by saying that $ A^3$ is a cube with edge length $ A^{1}$ again.

I wonder what the geometric interpration of $ \sqrt[2]{A}=A^{1/2}$ would look like. Is it a (straight) line? Is it constructible? Of course, we could extend this question by asking ourselves what $A^{k} $ would look like, in which $k \in \mathbb{R}$ or even $\mathbb{C}$ . When $k>2$, $A^k$ probably isn't constructible anymore on a sheet of paper, but one can still think about how these constructions of $A$ would 'look like'.

Thanks in advance,

Max Muller

P.S. I realize I ask more than one question now, which is an indictation I don't know a lot about this (yet) and I'd like to know more about this subject. Should this be a community wiki? Feel free to modify the Tags, I don't know how to classify this question exactly.