Question

How do you simplify $\sqrt{22.5}$ ?
 (a) Using Babylonian method
 (b) Using algebraic operations
 (c) Using linear operations
 (d) None of these

Answer 1

Hint: According to the question, we are to find the value of the given number $\sqrt{22.5}$. So, we can do it with a manual way of guessing the solution and reach the proper desired number. But now we can also see that the solution of the number is not a proper one and thus need to be calculated in some other way. So, we will try to use Babylonian method here.

Complete step by step solution:
We will start with dividing the number 22.5 by 2 to get the first guess for the square root .
First guess = $\dfrac{22.5}{2}$ = 11.25.
Next, we will go on with,
 Dividing 22.5 by the previous result. d = $\dfrac{22.5}{11.25}=2$.
 Average this value d with that of first step $\dfrac{2+11.25}{2}=6.625$which is the new guess.
 So, now to calculate the error = new guess - previous value = 11.25 - 6.625 = 4.625.
 Again, we see, 4.625 > 0.001. As error > accuracy, we repeat this step again.
Now, again dividing 22.5 by the previous result, d = $\dfrac{22.5}{6.625}$ = 3.3962264151.
We also have to average this value d with that of the previous step \[\dfrac{\left( 3.3962264151+6.625 \right)}{2}=5.0106132075\] , this is the new guess.
So, the error is now, new guess - previous value = 6.625 - 5.0106132075 = 1.6143867925.
and 1.6143867925 > 0.001. As again, error > accuracy, we repeat this step again.
Again,
 Dividing 22.5 by the previous result, d =\[\dfrac{~22.5}{5.0106132075}\] = 4.4904683455.
 Similarly, average this value d with that of previous step \[\dfrac{\left( 4.4904683455\text{ }+\text{ }5.0106132075 \right)}{2}\] = 4.7505407765
This is the new guess.
 Thus, Error = new guess - previous value = 5.0106132075 - 4.7505407765 = 0.260072431.
 But, 0.260072431 > 0.001. As error > accuracy, we repeat this step again.
Again, dividing 22.5 by the previous result, d =\[\dfrac{22.5}{4.7505407765\text{ }}=\text{ }4.7363028881\] .
Getting the average of this value d with that of previous step \[\dfrac{\left( 4.7363028881\text{ }+\text{ }4.7505407765 \right)}{2}=4.7434218323\]
This is the new guess.
Error = new guess - previous value = 4.7505407765 - 4.7434218323 = 0.0071189442. But, 0.0071189442 > 0.001. As error > accuracy, we will repeat this step again.
 Dividing 22.5 by the previous result, d =\[\dfrac{22.5}{4.7434218323}=4.7434111482\] .
 Averaging this value d with that of the last step,
 \[\dfrac{\left( 4.7434111482\text{ }+\text{ }4.7434218323 \right)}{2}=4.7434164903\]
 Error = new guess - previous value = 4.7434218323 - 4.7434164903 = 0.000005342.
 0.000005342 <= 0.001. As error <= accuracy, we stop the iterations and use 4.7434164903 as the square root.
Hence the solution is, (a) Using Babylonian method.

Note: The Babylonian method to find square roots is based on one of the numerical methods, which is based on the method for solving nonlinear equations. The idea is simple, starting from an arbitrary value of x, and y as 1, we can simply get the next approximation of root by finding the average of x and y. Then the y value will be updated with the number divided by x. And thus by checking the error we proceed further.