Calculators and computers use Newton's Method to compute square roots.
On this slide, we'll see how to compute $\sqrt{2}$.
Finding the square root of 2 is the same thing as solving $x^2 - 2 = 0$.
So we set $f(x)=x^2-2$, $f'(x)=2x$, and apply the recursive formula
$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} = x_n - \frac{x_n^2-2}{2x_n}.$$
It only takes a few steps to get 10 or 20 decimal places.