Newton's method is easy once you know how to apply it.
Find its derivative:
Now, apply the formula. Make an initial guess of, say, 0.5.
Do it again, using the last value:
It converges rather quickly if we choose a good initial guess.
Keep going if you wish, but that is pretty good.
See now what is going on?. They probably want you to keep going until your calculator goes as far as it can. With a guess of .5 that will be pretty fast. So continue.