How do you get the square root?

Answer 1

The answer depends on the number whose square root you wish to obtain and also what tools you have at your disposal. If you have a decent calculator or a computer then the solution is trivial.

If the number is a perfect square, then you need to factorise it. Pair off equal factors. Then the square root is the product of one from each pair. For example, to find sqrt(144):

144 = 2*2*2*2*3*3 = (2*2)*(2*2)*(3*3)

So sqrt(144) = 2 * 2 * 3 = 12.

Otherwise, you will have to use a numerical method, such as the Newton-Raphson method.

If you want to find the square root of kkk, define f(x) = x^2 – kkk.

Then finding the square root of kkk 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 want the square root of 7 and you you start with x0 = 5 (a pretty poor choice since 5^2 is 25, which is nowhere near 7).

Even so, x3 = 2.2362512515, which is less than 0.01% from the true value. Finally, remember that the negative value is also a square root.

Answer 2

Usually you can use a calculator.