Method for Extracting the $n$th root of a number?

Question

I was wondering if there exists a reliable way to extract the $n$th root of a given number. For example, if you have a large perfect square such as 10,404, how would I go about taking its square root by hand?

And what about cube roots? Is there a way to find the cube root of any number by hand?

If there is a way to take the cube and square roots of any number, is there a technique that can be applied to extract the cube or square root of polynomials? What about any root?

I think that, if there is a way to take the $n$th root of any number of polynomial, I would benefit from learning it, since I consider being able to work with roots important to my algebra foundation.

Thanks

Answer 1

If you would like to compute $\sqrt[p]{a}$, the following iterative process will do it to whatever accuracy you please:

First, make a guess and call it $x_0$. Then perform this iteration:

$$x_{n+1} = \frac{(p-1)x_n^p+a}{px^{k-1}}.$$

For instance, if you want $\sqrt[5]{11}$, guess that $x_0=2$. The iteration looks like

$$x_{n+1} = \frac{4x_n^5+11}{5x_n^4}.$$

So we have $$x_1 = \frac{4\cdot2^5+11}{5\cdot 2^4} = \frac{139}{80}=1.7375.$$

$$x_2=\frac{4\cdot1.7375^5+11}{5\cdot 1.7375^4} = 1.631392308.$$

This number to the 5th power is $11.55558806$, so we're getting close.

$$x_3=\frac{4\cdot1.631392308^5+11}{5\cdot 1.631392308^4} = 1.615704970.$$

And $1.615704970^5 = 11.0105\ldots.$

This is an implementation of Newton's Method on the polynomial $x^p-a$.