Question

What is the square root of \[110.25\]?

Answer 1

Hint: In order to determine the square root of the given decimal number \[110.25\] which is a positive number.
So, the solution can be found in the two process as
> Long division method or
> Babylonian method or hero’s method
For this question the Babylonian method is made to apply by finding out the accuracy which is equalized to its error value.

Complete step by step solution:
In this case we are going to use the ‘Babylonian method’ to get the square root of any positive number.
We must set an error for the final result.
In other words, we have to correct it to at least \[2\] decimal places.
Step 1:
Divide the number (110.25) by 2 to get the first guess for the square root.
First guess = \[\dfrac{{110.25}}{2} = 55.125\]
Step 2:
Divide 110.25 by the previous result.
\[d = \dfrac{{110.25}}{{55.125}} = 2\]
Average this value (d) with that of step 1:
 \[\dfrac{{(2 + 55.125)}}{2} = 28.5625\] (new guess).
Error \[ = \] new guess \[ - \] previous value\[ = 55.125 - 26.5625 = 26.5625\].
\[26.5625 > 0.001\].
As error \[ > \] accuracy. The step is repeated again.
Step 3:
Divide 110.25 by the previous result.
\[d = \dfrac{{110.25}}{{28.5625}} = 3.8599562363\]
Average this value (d) with that of step 2:
\[\dfrac{{(3.8599562363 + 28.5625)}}{2} = 16.2112281182\](new guess)
Error \[ = \] new guess \[ - \] previous value \[ = 28.5625 - 16.2112281182 = 12.3512718818\]
\[12.3512718818 > 0.001\].
As error \[ > \] accuracy. The step is repeated again.
Step 4:
Divide \[110.25\] by the previous result.
\[d = \dfrac{{110.25}}{{16.2112281182}} = 6.8008419347\]
Average this value (d) with that of step 3:
\[\dfrac{{(6.8008419347 + 16.2112281182)}}{2} = 11.5060350265\](new guess)
Error \[ = \] new guess \[ - \] previous value \[ = 16.2112281182-11.5060350265 = 4.7051930917\].
\[4.7051930917{\text{ }} > {\text{ }}0.001\].
As error \[ > \] accuracy. The step is repeated again.
Step 5:
Divide 110.25 by the previous result.
\[d = \dfrac{{110.25}}{{11.5060350265}} = 9.5819280704\]
Average this value (d) with that of step 4:
\[\dfrac{{(9.581819280704 + 11.50606350265)}}{2} = 10.5439815484\](new guess)
Error \[ = \] new guess \[ - \] previous value \[ = 11.5060350265-10.5439815484 = 0.9620534781\].
\[0.9620534781 > 0.001\].
As error \[ > \] accuracy. The step is repeated again.
Step 6:
Divide 110.25 by the previous result.
\[d = \dfrac{{110.25}}{{10.5439815484}} = 10.4562019095\]
Average this value (d) with that of step 5:
\[\dfrac{{(10.4562019095 + 10.5439815484)}}{2} = 10.500091729\] (new guess)
Error \[ = \] new guess \[ - \] previous value \[ = {\text{ }}10.5439815484{\text{ }}-{\text{ }}10.500091729{\text{ }} = {\text{ }}0.0438898194\]
\[0.0438898194\; > 0.001\].
As error \[ > \] accuracy. The step is repeated again.
Step 7:
Divide 110.25 by the previous result.
\[d = \dfrac{{110.25}}{{10.500091729}} = 10.4999082718\]
Average this value (d) with that of step 6:
\[\dfrac{{(10.4999082718 + 10.500091729)}}{2} = 10.5000000004\] (new guess)
Error \[ = \] new guess \[ - \] previous value \[ = 10.500091729-10.5000000004 = 0.0000917286\].
\[0.0000917286 \leqslant 0.001\].
As error \[ \leqslant \] accuracy.
Thus, we stop the iterations and use \[10.5000000004\] as the square root.
Therefore, the square root of \[110.25 = 10.5\].

Note:
The diagonal length of a square with side lengths of \[1\] equals the square root of \[2\]. The diagonal of a cube with side length \[ = 1\] is the square root of \[3\].
A perfect square is the product of a number multiplied by itself. The square root of the numbers multiplied to get the perfect square. This implies that perfect squares can only be created by multiplying rational (whole) numbers.
For certain numbers, the square root is irrational (not a whole integer). This makes it difficult to measure in your mind, but you can find the square root of any number using a calculator or a map.