How do you calculate square root of 156.25?

Answer 1

You can make a rough estimate by squaring numbers. 10x10 is 100. 11x11 is 121, 12x12 is 144, 13x13 is 169. So it is approximately 12.5.

The best way is to get a calculator and hit the tick sign and then type in 156.25. This gives you 12.5, so my estimate actually got it right!

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The following are two more systematic ways:

The first is to express 156.25 as a top heavy fraction.

156.25 = 1561/4 = 625/4.

Then, sqrt(156.25) = sqrt(625/4) = sqrt(625)/sqrt(4) = 25/4 = 12.5

That method relies on your knowing that 156.25 has a rational square root. If you did not know that, perhaps the simplest method is an iterative method - of which the Newton Raphson is one. This method entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage.

To start with, if you want to find the square root of 156.25, define f(x) = x2 - 156.25.

Then finding the square root of 156.25 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. Even if your first guess is not so good:

You know that the square root of 100 is 10.

So suppose you start with x0 = 10 (not the best possible choice since 102 is less than 2/3 of your target).

Even so, x3 = 12.000006, approx, an error of less one in ten trillion.

Finally, remember that the negative value is also a square root.

Answer 2

√156.25