[Trantor]

It seems to me whatever manipulation you do along the lines you are trying will always retain terms of the form sqrt(a +/- sqrt(b)). That means you won't be able to cancel out terms to get to the answer. The equation you first mentions works to simplify the collection of term because it reduces them all to the same type , ie sqrt(x), namely the type of term you are trying to end up with.

Thus it seems to me your approach can't work. Don't know if this helps but I've convinced myself there is no point in trying for the type of approach you are tryng.