> I have always thought that sqrt(a/b) and sqrt(a)/sqrt(b) are > exactly the same >
Certainly sqrt(a/b) and sqrt(a)/sqrt(b) are both square roots of a/b. But a complex number has two square roots. It could happen (as Axel explains), that the "principal branch" choice for sqrt results in opposite choices for these two. If a,b are both positive, this does not happen, and you get the same square root.
Similar things can happen with other powers, with logarithgms, inverse trig functions, and so on. Unless you choose the arguments nicely enough.