The calculus approach is to use Newtons Method:
[tex]x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}[/tex]
where
[tex]f(x) = x^2 - a[/tex]
Start with initial guess of "a".
After 4 or 5 iterations, you should get a close approximation to [tex]\sqrt{a}[/tex]
...............
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.
[tex]m^2\ \textless \ a \ \textless \ n^2[/tex]
Then use midpoint [tex]\frac{m+n}{2} [/tex] for next guess, if
[tex](\frac{m+n}{2})^2 \ \textless \ a[/tex]
then repeat with "m" = midpoint. Otherwise repeat with "n" = midpoint.
Continue until you have a good approximation, about 6-8 decimal points.
