Question

Find the number whose square is the given number 14641.

Answer 1

Hint: It is given that we have to find the number whose square is the given number 14641 which means we have to find the square root of 14641. You may find the square root of 14641 by prime factorization but it will take a lot of time to do that so here we are using a long division method to find the square root of 14641.

Complete step-by-step answer:
The number which we have to find the square root of:
14641
We are going to use the long division method to find the square root of 14641.
In the long division method, starting from the first number 1 as you can see 1 is the square of 1 so we divide 1 by 1 as follows:
 $ 1\left| \begin{align}
  & \underline{1}4641 \\
 & \dfrac{1}{04} \\
\end{align} \right|1 $
Now, carry the remaining two digit number from 14641 which will be 46 and then add 1 into the number 1 in which we divide the number 14641 so adding 1 into 1 we get 2 and by hit and trial method write a number after 2 in such a way that multiplying the same number with the new number will give the number which is less than or equal to 46 so writing 2 after 2 which makes the number 22 and then multiplying 2 by 22 we get 44 which is less than 46.
\[\begin{matrix}
   1 \\
   22 \\
\end{matrix}\left| \dfrac{\begin{align}
  & \underline{1}4641 \\
 & \dfrac{1}{0} \\
\end{align}}{\begin{align}
  & 46 \\
 & \dfrac{44}{02} \\
\end{align}} \right|12\]
Now, carry the remaining part of the number i.e. 41 and add 2 to 22 which makes 24 so write a number after 24 in such a way that multiplying the same number with the number will give the answer which is less than or equal to 241 so writing 1 after 24 then number becomes 241 and multiplying 1 by 241 will give 241.
\[\begin{matrix}
   1 \\
   1 \\
   22 \\
   2 \\
   241 \\
\end{matrix}\left| \dfrac{\begin{align}
  & \underline{1}4641 \\
 & \dfrac{1}{0} \\
\end{align}}{\begin{align}
  & 46 \\
 & \dfrac{44}{\begin{align}
  & 0241 \\
 & \dfrac{0241}{0000} \\
\end{align}} \\
\end{align}} \right|121\]
Hence, the number whose square is 14641 is 121.

Note: You can verify the square root that we are getting above by taking the square of 121 and see whether the square is 14641 or not.
Multiplying 121 by itself we get,
 $ \begin{matrix}
  \text{ }121 \\
  \dfrac{\times 121}{\dfrac{\begin{matrix}
   121 \\
   242\times \\
   121\times \times \\
\end{matrix}}{14641}} \\
\end{matrix} $
From the above multiplication of 121 by itself we are getting the result as 14641 which is the same as the number given in the question.
Hence, the square root that we have got from the long division method is correct.