Question

How do I approximate a square root

Answer 1

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.