How do i approximate a square root

The calculus approach is to use Newtons Method:

where 

Start with initial guess of "a".
After 4 or 5 iterations, you should get a close approximation to 
...............

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.

Then use midpoint  for next guess, if 

then repeat with "m" = midpoint. Otherwise repeat with "n" = midpoint.
Continue until you have a good approximation, about 6-8 decimal points.