You could use an iterative numerical method. Suppose you want to find the square root of the number m, say.
Makle an initial guess x0 (just set it to m/4 or something like it) then for a number of iterations i, use
xi = (xi-1 + m/xi-1)/2 until xi is the same as xi-1.
For example, suppose you want to find the square root of 3: