Hello Wolfgang,
By default Mathematica only performs a "small" set of transformations on expressions. To get the result you want, you can use the function Simplify.
In:= Simplify[x y^2/x^-3*(x y^-2/(x (y^2)^0) - 3 y^-2/x^4)]
Out= -3 + x^4
Simplify performs several transformations on expressions and returns the simplest form. For more complex expressions, it can happen that Simplify will not lead to the result sought. In such cases, one can use the function FullSimplify. With the latter, a larger number of transformations will be tried. For instance for an expression with the Euler Gamma function:
In:= Simplify[x Gamma[x]]
Out= x Gamma[x]
In:= FullSimplify[x Gamma[x]]
Out= Gamma[1 + x]
