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.
Chat with our AI personalities
Press the square root button on your calculator.
The square root of a/b is equal to the square root of a divided by the square root of b. I hope this helps you.
The square root of 289 is ± 17.
The square root of 1225 is 35.
The square root of 8649 is 93.