## Cubic Root

I still remember how to extract a square root without a computer but could somebody remind me the technique to find a cubic root just with the pencil and paper?
 Recognitions: Gold Member Homework Help Science Advisor I would suggest using some Newton-Raphson scheme. 1. Let $$f(x)=x^{3}-a$$ You are to find X so that f(X)=0. 2. Pick an initial value $$x_{0}\to{f}(x_{0})=x_{0}^{3}-a$$ 3. The equation for the tangent line L(x)=at $$(x_{0},f(x_{0})$$ is given by: $$L(x)=f(x_{0})+f'(x_{0})(x-x_{0})$$ 4- Let the next iteration point be the x-intercept of L(x): $$L(x_{1})=0\to{x}_{1}=x_{0}-\frac{f(x_{0})}{f'(x_{0}}$$ 5. Or, in this case, the iterative scheme becomes: $$x_{n}=x_{n-1}-\frac{x_{n-1}-\frac{a}{x_{n-1}^{2}}}{3}$$ That is: $$x_{n}=\frac{2x_{n-1}^{3}+a}{3x_{n-1}^{2}}, n\geq{1}$$
 Recognitions: Homework Help Science Advisor I'm actually old enough to remember this. It's been somewhat wisely forgotten. http://www.nist.gov/dads/HTML/cubeRoot.html. You may wish to also check out the Isaac Asimov story, "The Feeling of Power". Kind of haunting, these days.

## Cubic Root

 Quote by Dick I'm actually old enough to remember this. It's been somewhat wisely forgotten. http://www.nist.gov/dads/HTML/cubeRoot.html. You may wish to also check out the Isaac Asimov story, "The Feeling of Power". Kind of haunting, these days.
Thanks, now it is coming back!!