Show how you find the square root of 8?

Answer 1

You could use the Newton-Raphson method.

To find the square root of 8, define f(x) = x^2 – 8. Then finding the square root of 8 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 = 2.5 (a pretty poor choice since 5^2 is 25, which is nowhere near 7).
Even so, x3 = 2.8284271259, which is around a billionth from the true value. Finally, remember that the negative value is also a square root.However, you may need a calculator for doing the iteration. And, if you have a calculator, you may as well use the square root key!

There is another method which resembles long division but the graphic facility of this browser is hopelessly inadequate for showing that method.

Finally, there is bracketing the square.

2^2 = 4 < 8 < 0 = 3^2 so 2 < sqrt(8) < 3

Next try 800, whose sqrt will lie between 2*10 and 3*10.

28^2 = 784 < 800 < 841 = 29^2 so 2.8 < sqrt(8) < 2.9

Next go for 800*100, with the number bracketed in 280 - 290, and so on. This will require some tedious calculations but can be done without a calculator.