Square root formula

Answer 1

If you have a scientific calculator then it will have a radical key for finding the square root of a number. In a spreadsheet it is sqrt(x) = x^0.5 or x^(0.5).

If you do not have a calculator or computer, then there is no simple formula. 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 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.