Suppose I have a smooth algebraic surface $X$ and a subscheme $Z$ such that the blowup $$Y = Bl(X,Z)$$ is smooth.

Now $Z$ does not have to be smooth, say if it is given by some power of the maximal ideal at a point, but is it nevertheless the case that $Y$ is isomorphic to the iterated blowup of $X$ along smooth points?