Question

What is the square root of 145?

Answer 1

Hint: In the above problem, we are asked to find the square root of 145 and which we are going to find by dividing the number into two parts: the first part contains the first two numbers (145) and the second part contains the last two numbers (145). Then we will find a number by hit and trial whose square is lesser than or equal to the first number (i.e. 1). Then we subtract the square of the number obtained from hit and trial from the first number and write the remaining number. Remaining steps will be more clear in the below solution.

Complete step by step answer:
The number which we are asked to find the square root is:
145
We are going to find the square root of 145 by division method.
Now, we are going to divide the number into two parts in the following manner.
1 45
By hit and trial method, the number whose square is equal to 1 is 1. We know that the square of 1 is 1 and which is equal to 1.
\[1\left| \begin{align}
  & \underline{1}45 \\
 & 1 \\
 & \overline{0} \\
\end{align} \right|1\]
In the above division method, we have subtracted 1 from 1. Now, we are going to write the remaining number 45 after 0 and then the above division will look like:


\[1\left| \begin{align}
  & 145 \\
 & 1\downarrow \\
 & \overline{045} \\
\end{align} \right|1\]
Now, we are going to multiply 1 by 2 and then write 2 below 1.
\[\begin{matrix}
   1 \\
   22 \\
\end{matrix}\left| \begin{align}
  & 145 \\
 & 1\downarrow \\
 & \overline{\begin{align}
  & 045 \\
 & \dfrac{044}{001} \\
\end{align}} \\
\end{align} \right|12\]
Now, we are going to add 2 to 22 then we get 24 and as we have put the decimal so one zero is written after 1 followed by one zero which will come from 240. Similar process will be repeated till will get the square root of 145.
\[\begin{matrix}
   1 \\
   22 \\
   240 \\
   2404 \\
   24081 \\
\end{matrix}\left| \begin{align}
  & 145.000000 \\
 & 1\downarrow \\
 & \overline{\begin{align}
  & 045 \\
 & 044 \\
 & \overline{\begin{align}
  & 00100 \\
 & 00000 \\
 & \overline{\begin{align}
  & 0010000 \\
 & 0009616 \\
 & \overline{\begin{align}
  & 000038400 \\
 & 000024081 \\
 & \overline{000014319} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align} \right|12.041\]

From the above long division method, we have come to a conclusion that the square root of 145 is 12.041.

Note: You can check whether the square root you have found is correct or not by multiplying the square root of 145 by itself and see if we are getting the same number which we have started with.
The square root of 145 which we have calculated above is equal to 12.041 so multiplying 12.041 by 12.041 we get,
$\begin{align}
  & 12.041\times 12.041 \\
 & =144.98 \\
\end{align}$
Rounding off the above number will give us 145.
Hence, the square root which we have found above is correct.