I guess it's the binomial expansion that is giving you problems rosala.
Another way of approaching this is to use the following, iterative formula for square roots (I'll just present it without deriving it):
If we have x2 = N then
xn+1 = (1/2)(xn + N/xn)
where n is iteration number.
So, if N = 47 and we start with, say, x0 = 6
then x1 = (1/2)(6 +47/6) = 6.917 which is a first approximation
x2 = (1/2)(6.917 + 47/6.917) = 6.856 which is a better approximation.
If we start with x0 = 7
then x1 = (1/2)(7 + 47/7) = 6.857 which is a better first approximation.
Of course, this depends on your belief in the iterative formula xn+1 = (1/2)(xn + N/xn). I have just stated it rather than deriving it because the derivation involves calculus, which you probably haven't done yet (actually, there might be a non-calculus derivation, but I can't think of it off-hand!).
I don't know if this helps any!
Also, the level of precision of the result is not specified in the question, so it could be that just 7 is an acceptable approximation!
