Show that Newton's method applied to (where n and c are positive constants) produces the iterative scheme for approximating . We learned Newton's method, but we haven't applied it to "schemes" and my teacher told us to look at this problem tonight. Can someone explain, please! Thanks.

I'm inept with Latex, so I apologize for the mess this may be.
Newton's method, you might remember, takes some guess say "a" as a zero for a function and makes a better approximation so long as the method doesn't break down (which it can). So the formula for this better approximation say "b", b=a-(f(a)/f'(a)).
So what your teacher gave you is another form of this approximation "b" being x(n+1) in your formula for a certain case "x^n-c=0". So b=a-(a^(n)-c/na^(n-1)). So at this point a is x(n) in your formula and a little algebra will give your formula. Start with a common denominator on the RHS, then divide each term out by the denominator. Then factor out a (1/n) and voila, there you have it. Once again sorry about not using Latex, I ought to learn how to use it.