0
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
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