What is the method for finding the square root of 89?

Answer 1

Well, there are various ways. The most simple way is to approximate the solution:

89 is between 9^2 and 10^2, therefore the first digit is 9

89 is between 9.4^2 and 9.5^2, therefore the second digit is 4

There is a second method. in which you use the Newton's method:

Consider the following equation: f(x)=x^2-89, which has sqrt[89] as its root.

Let x0 (the starting value) be 9.

This is the formula of Newton's method:

x_(n+1)= x_(n)- f(x_n)/f'(x_n)

x_(n+1)= x_(n)- (x_(n)^2-89)/2*(x_n)

Now we begin the approxiamton:

x_(2)= 9- (9^2-89)/(2*9)=9.4444

x_3=9.433

x_3=9.43398132

If you need digits, just use more iterations (The numbers of digits doubles for each iteration ).