How do you get the square root of 709?

Answer 1

One way to do this would be with Newton's method:

Let:

f(x) = 709 - x2

f'(x) = -2x

We'll start with 27 as an approximation:

x0 = 27

x1 = x0 - f(x0) / f'(x0) = 27 - (709 - 729) / -2(27) = 27 - 10/27 ≈ 26.629629

x2 = x1 - f(x1) / f'(x1) ≈ 26.629629 - (709 - 709.137174) / -2(26.629629) ≈ 26.627053

x3 = x2 - f(x2) / f'(x2) ≈ 26.627053 - (709 - 708.999951) / -2(26.627053) ≈ 26.627053

So we know that the square root is approximately 26.627053. You can check this of course by squaring our answer:

26.6270532 = 708.999951