How do I take the root of a decimal?

Answer 1

There is no simple or general method.

One option is to convert the decimal number to a rational fraction - proper or improper.

Then sqrt(x/y) -= sqrtx)/sqrt(y) may provide a solution.

For example,

s = sqrt(0.87890625) = sqrt(87890625/100000000)

GCF(87890625, 100000000) = 78125

therefore, s = sqrt(225/256) = sqrt(225/sqrt(256) = 15/16.

This only works if x/y is a square of some rational fraction. If not, the Newton Raphson method is best.

If you want to find the square root of kkk, define f(x) = x^2 – kkk. Then finding the square root of kkk 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 7.7 and you you start with x0 = 3 (a pretty poor choice since 3^2 is 9, which is not exactly 7.7).
Even so, x3 = 2.774887385, which is less than a billionth of 1% from the true value. Finally, remember that the negative value is also a square root.