I understand that the addition operator, defined as $F(x,y)=x+y$, and the multiplication operator, defined as $F(x,y)=xy$, are both commutative and associative (meaning, of course, that $F(y,x)=F(x,y)$ and that $F(F(x,y),z)=F(x,F(y,z))=F(F(x,z),y)$). My question is: Why are exponentiation, tetration, pentation, etc. not generally commutative or associative? I understand that exponentiation just works that way, but my question is, is there an intuitive explanation and its mathematical proof?
Cheers, and thanks in advance.
I hope I've explained everything like I intended to. If I messed something up, feel free to edit or redirect me to a different post.