Question

Find the value of \[\sqrt{784}\] by using the long division method.

Answer 1

Hint: We solve this problem by using the long division method of 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 the 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 \[\sqrt{784}\] 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 the 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 784 as the pairs of 2 numbers starting from right then we get
\[\Rightarrow \left| \!{\overline {\,
 784 \,}} \right. \]
Here we can see that the pairs are 7, 84
Now let us take the perfect square less than and near to 7 that is 4
By taking the square root of 4 that is 2 as first digit of required square root then we get
\[\Rightarrow \begin{matrix}
   2 \\
   2\left| \!{\overline {\,
 \begin{align}
  & 784 \\
 & -4 \\
 & =384 \\
\end{align} \,}} \right. \\
\end{matrix}\]
Now, by subtracting number 7 and 4 and taking the second pair 84 then we get
\[\Rightarrow \begin{matrix}
   2 \\
   \left| \!{\overline {\,
 384 \,}} \right. \\
\end{matrix}\]
Now, let us double the number 2 to form the equation as
\[4\underset{\scriptscriptstyle-}{k}\times \underset{\scriptscriptstyle-}{k}\le 384\]
Here, we know that \['k'\] is a digit.
By, using the trial and error method we get
\[48\times 8=384\]
By using this condition we get
\[\Rightarrow \begin{matrix}
   28 \\
   48\left| \!{\overline {\,
 \begin{align}
  & 384 \\
 & -384 \\
 & =0 \\
\end{align} \,}} \right. \\
\end{matrix}\]
Here, we can see that we got 0 so that we can stop the process.

Therefore we can conclude that the square root of 784 is 28.

Note: Students may make 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 mistake and do the pairs from left side and take the pairs as 78 and 4
This gives the wrong answer.