Suppose I have two expressions; call them $A$ and $B$. The following values of $A$ and $B$ are good examples for my question...
$$ A = \pi e^2\\ B = \pi^2 e $$
Is there a method to determine the truth of comparisons such as $A \ge B$, $A \lt B$, etc? Especially, I want to ensure that exactness is preserved if possible, so I'd like to avoid converting $A$ and $B$ to numbers.
Actually, I believe I can be a bit more specific. Even if the number is the result of using integers, addition, subtraction, multiplication, division, and the placement of square root signs anywhere, there is no canonical way of writing this number. Meaning there is no evident way of telling whether it is positive or negative: a certain amount of cleverness in squaring sides, gathering on one side, etc. will be required. Thus, if we take the square root of what we have so far, we cannot entirely be sure the result is real.