How to find recurrence relation for $f(x)=x^3-\alpha=0$ where $\alpha>0$ using newton raphson method?

Question

Alright for $x^3-\alpha=0$ we know that $x=\sqrt[3]\alpha$ is the root of course but unlike defined number it is too complicated to be solved using $x_0=\sqrt\alpha$ or $x_0=1$ for example so what is the idea behind it?

Answer 1

If $f(x)=x^3-\alpha$ then the Newton Raphson method would give the recurrence relation $$x_{n+1}=x_n-\frac{x_n^3-\alpha}{3x_n^2}$$