How do you find sq root of 65?

Answer 1

One way to find the square root of a number is an iterative method. This entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage. One such is the Newton-Raphson method.

To start with, if you want to find the square root of 65, define f(x) = x2 - 65.

Then finding the square root of 65 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, …

After a few iterations, xn will be very close to the required square root.

Now 82 = 64 so 8 = sqrt(64). Therefore 8 is a pretty good place to start for sqrt(65). That then gives x2 = 8.62258 which is less than 4 billionths away from the true value. Remember also that -8.62258 is a square root.