Newton-Raphson's Method to find $\sqrt{2012}$

Question

I am asked to find $\sqrt{2012}$ using Newton-Raphson's Method with the following recursive method

$$x_{n+1} = \frac{1}{2} (x_n + \frac{a}{x_n}) $$

I notied that give same answers as using

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$

This is easy, but the next part asks to find a similar recursive method to find $\sqrt[3]{2012}$. How do I find such a method?

UPDATE

I did

$$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 12.7551$$

$$x_2 = 12.6257$$

$$x_3 = 12.6244$$

$$x_4 = 12.6244$$

$$x_4 ^ 3 = 2012.02$$

Which seems correct. But I didn't use a similar recursive method like the question asked?

Answer 1

Hint: What does Newton's method say for $f(x) = x^3 - a$?