How do you find the square root of 597?

Answer 1

The simplest method is to use the square root key on a calculator. But assuming you cannot do that:

One alternative is the Newton-Raphson method. The process for using the method is described below, you can find out more about the rationale of the N-R method if you look for Newton's method on Wikipedia.

Define f(x) = x2 - 597

Its derivative, f'(x) = 2x

Make a guess at sqrt(597), say x0.

Calculate xn+1 = xn + f(xn)/f'(xn) for n = 0, 1, 2 etc.

Even with an outrageously high starting value, x0, of 30, the second iteration x2, has an error of around 2 in a hundred thousand!

Another option is bracketing.

Find two integers such that their squares bracket 597

242 = 576 < 597 < 625 = 252 so 24 < sqrt(597) < 25

Next find two numbers with 1 decimal place whose squares bracket 597

24.42 = 595.36 < 597 < 600.25 = 24.52 so 24.4 < sqrt(597) < 24.5

And so on, until you reach a satisfactory level of precision.

Finally, there is a method that resembles long division but this browser is not suitable for describing that method.