The ancient Greeks knew how to efficiently compute square roots. If you want the square root of a and already have a guess x_old for it, then you can have a better guess as the average of x_old and a / x_old, i.e.

x_new = (x_old + a / x_old) x 0.5

For the initial guess, set x_old to a.

Implement a program that reads a, and n, and that prints all guesses other than the initial one, until two of them are less than 10^-n apart. Use BigDecimal.

For the division, use

divide(xold, n + 1, RoundingMode.HALF_EVEN)