Question

Find the square root of the following by long division method.
a) $1369$
b) $5625$

Answer 1

Hint: From the question we have to find the square root of the $1369$ and $5625$ by long division method. Long division method is used to find the square root of any number. We will see in the solution below how to do the long division method.

Complete step by step solution:
Firstly, we will solve a
a) $1369$
Taking $1369$ as the number whose square root is to be evaluated by using a long division method. Place a bar over the pair of numbers starting from the unit place or Right-hand side of the number. In case, if we have the total number of digits as odd number, the leftmost digit will also have a bar, here in the number there are even number of digits so,
$\Rightarrow \overline{13}\ \overline{69}$
Now, we have to take the largest number as the divisor whose square is less than or equal to the number on the extreme left of the number. The digit on the extreme left is the dividend. Divide and write the quotient. Here the quotient is $3$ and the remainder is $4$
Here the largest number is $3$.
\[\begin{align}
  & \text{ 3} \\
 & \text{ 3}\left| \!{\overline {\,
 \begin{align}
  & \underline{\begin{align}
  & \overline{13}\ \overline{69} \\
 & 9 \\
\end{align}} \\
 & 4 \\
\end{align} \,}} \right. \\
\end{align}\]
Next, we then bring down the number, which is under the bar, to the right side of the remainder. Here, in this case, we bring down $69$. And also, we have remainder $4$. Now, $469$ is our new dividend.
Now double the value of the quotient and enter it with blank space on the right side. That is $6$.
Again, we have to select the largest digit for the unit place of the divisor $\left( 6\ldots \right)$ such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than the dividend ($469$).
In this case, $67\times 7=469$. So, the new digit is $7$.
\[\begin{align}
  & \text{ 37} \\
 & \text{ 67}\left| \!{\overline {\,
 \begin{align}
  & \underline{\begin{align}
  & \overline{13}\ \overline{69} \\
 & 9 \\
\end{align}} \\
 & \underline{\begin{align}
  & 469 \\
 & 469 \\
\end{align}} \\
 & 0 \\
\end{align} \,}} \right. \\
\end{align}\]
The remainder is 0, and we have no number left for division
Therefore $\sqrt{1369}=37$.
Now we will solve for b
 b) $5625$
similarly, for b also we will do as we did in a.
by following those steps, we will get
\[\begin{align}
  & \text{ 7} \\
 & \text{ 7}\left| \!{\overline {\,
 \begin{align}
  & \underline{\begin{align}
  & \overline{56}\ \overline{25} \\
 & 49 \\
\end{align}} \\
 & 7 \\
\end{align} \,}} \right. \\
\end{align}\]
\[\begin{align}
  & \text{ 75} \\
 & \text{ 145}\left| \!{\overline {\,
 \begin{align}
  & \underline{\begin{align}
  & \overline{56}\ \overline{25} \\
 & 49 \\
\end{align}} \\
 & \underline{\begin{align}
  & 725 \\
 & 725 \\
\end{align}} \\
 & 0 \\
\end{align} \,}} \right. \\
\end{align}\]
The remainder is 0, and we have no number left for division
Therefore $\sqrt{5625}=75$.

Note: Students should know the long division method to find the square root of the number. Students should guess the correct number while dividing in the question “a” the square of $2$ is also a lesser than the $13$ but we should select the largest number whose square should be lesser than $13$ if we proceed with $2$ the whole answer will be wrong.