Question

Find the value of square root of 2304 by using the long division method.

Answer 1

Hint: We solve this problem by using the long division method of the square root.
The steps involved in finding the square root using the long division method are shown below:
(1) We divide the number into pairs of 2 digits starting from the right side.
(2) We take the square less than and near to first pair on the left and take the square root of that near number as the first digit of the required value. Let us assume it as \['x'\]
(3) We subtract the first pair and the near square to that pair and take the second pair as a set of numbers.
(4) If the new set of number is \['n'\] then we double the first digit we got in the step (2) to make the problem as
\[2x\underset{\scriptscriptstyle-}{k}\times \underset{\scriptscriptstyle-}{k}\le n\]
Here \[2x\underset{\scriptscriptstyle-}{k}\] is a number but not \[2x\times k\]
Here, the value of \['k'\] found by train and error method will be the second digit of the required square root.

Complete step by step answer:
We are asked to find the value of \[\sqrt{2304}\]by using the long division method.
We know that the steps involved in this process are
(1) We divide the number into pairs of 2 digits starting from the right side.
(2) We take the square less than and near to first pair on the left and take the square root of that near number as the first digit of the required value. Let us assume it as \['x'\]
(3) We subtract the first pair and the near square to that pair and take the second pair as a set of numbers.
(4) If the new set of number is \['n'\] then we double the first digit we got in the step (2) to make the problem as
\[2x\underset{\scriptscriptstyle-}{k}\times \underset{\scriptscriptstyle-}{k}\le n\]
Here \[2x\underset{\scriptscriptstyle-}{k}\] is a number but not \[2x\times k\]
Here, the value of \['k'\] will be the second digit of the required square root.
Let us take the number 2304 as the pairs of 2 numbers starting from right then we get
\[\Rightarrow \left| \!{\overline {\,
 \overline{23}\overline{04} \,}} \right. \]
Here we can see that the pairs are 23, 04
Now let us take the perfect square less than and near to 23 that is 16
By taking the square root of 16 that is 4 as the first digit of the required square root then we get
\[\Rightarrow \begin{matrix}
   4 \\
   4\left| \!{\overline {\,
 \begin{align}
  & \overline{23}\overline{04} \\
 & -16 \\
 & =704 \\
\end{align} \,}} \right. \\
\end{matrix}\]
Now, by subtracting number 23 and 16 and taking the second pair 04 then we get
\[\Rightarrow \begin{matrix}
   4 \\
   \left| \!{\overline {\,
 704 \,}} \right. \\
\end{matrix}\]
Now, let us double the number 4 to form the equation as
\[8\underset{\scriptscriptstyle-}{k}\times \underset{\scriptscriptstyle-}{k}\le 704\]
Here, we know that \['k'\] is a digit.
By, using the trial and error method we get
\[88\times 8=704\]
By using this condition we get
\[\Rightarrow \begin{matrix}
   48 \\
   88\left| \!{\overline {\,
 \begin{align}
  & 704 \\
 & -704 \\
 & =0 \\
\end{align} \,}} \right. \\
\end{matrix}\]
Here, we can see that we got 0 so that we can stop the process.
Therefore we can conclude that the square root of 2304 is 48.

Note:
Students may do mistakes in taking the pairs of the numbers.
We have the first step of the long division process that is,
(1) We divide the number into pairs of 2 digits starting from the right side.
Here, it says that we need to do pairs from the right side. So, the pairs for the number 784 will be 7 and 84
But students may do mistakes and do the pairs from the left side and take the pairs like 78 and 4 This gives the wrong answer.