How do you get the decimal expansion of the square root of 28?

Answer 1

Since sqrt(28) is an irrational number you can only get an approximate decimal expansion.

There are many numerical techniques: a simple one to use is the Newton-Raphson method. This entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage.

If you want to find the square root of 28, define f(x) = x2 - 28.

Then finding the square root of 28 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 reasonable choice since 52 is 25, which is pretty close to 28).

Then so, x1 = 5.2915094340, which is less than 1% from the true value.

x2 = 5.3, which is an error of around 1 in 150,000 and by the next iteration the error is less than 5 in a trillion.

Finally, remember that the negative value is also a square root.