Question

How do I approximate a square root

Answer 1

The calculus approach is to use Newtons Method:

`x_(n+1) = x_n - (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
`√(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
`(m+n)/(2)` for next guess, if

`((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.