The calculus approach is to use Newtons Method:
\(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)
where
\(f(x) = x^2 - a\)
Start with initial guess of "a".
After 4 or 5 iterations, you should get a close approximation to \(\sqrt{a}\)
...............
If you wanted a non-calculus approach, I would suggest divide and conquer.
First determine which 2 integers the sqrt lies in between by perfect squares.
\(m^2\ \textless \ a \ \textless \ n^2\)
Then use midpoint \(\frac{m+n}{2}\) for next guess, if
\((\frac{m+n}{2})^2 \ \textless \ a\)
then repeat with "m" = midpoint. Otherwise repeat with "n" = midpoint.
Continue until you have a good approximation, about 6-8 decimal points.
