How do you find square root on a simple calculator?

Without a square root key, there is no simple way to find square roots. The best method is probably the Newton Raphson iteration.

If you want to find the square root of k, define f(x) = x^2 – k. Then finding the square root of k is equivalent to solving f(x) = 0.

Let f’(x) = 2x. This is the derivative of f(x) but you do not need to know that to use the N-R method.

Start with x0 as the first guess. Then let xn+1 = xn - f(xn)/f’(xn) for n = 0, 1, 2, …
Provided you made a reasonable choice for the starting point, the iteration will very quickly converge to the true answer. It works even if your first guess is not so good:
Suppose you want the square root of 7 and you start with x0 = 5 (a pretty poor choice since 5^2 is 25, which is nowhere near 7).
Even so, x3 = 2.2362512515, which is less than 0.01% from the true value. Finally, remember that the negative value is also a square root.