Question

What is the square root of $7225$ using the long division method?

Answer 1

Hint: Square root of a number is a number that when multiplied with itself yields the original number. In the given question, we need to find the square root of the given number using the long division method. There are various steps involved in the process of calculating the square root of a number using the long division process as discussed in the solution.

Complete step by step solution:
We will find the square root of the number $7225$ using the long division method.
Steps to find square root using long division:
Step 1: Write down the number whose square root is to be found and place a bar over the pairs of two numbers, starting from the left side.
\[7225\]

Step 2: For the divisor, take the largest number whose square is less than or equal to the first pair of numbers. Here, the first pair is $72$. So, $8$ is the largest number whose square is less than or equal to $72$. So, we will divide by $8$.
\[\begin{align}
  & \,\,\,\,\,8 \\
 & 8\left| \!{\overline {\,
 \begin{align}
  & \overline{72}\overline{25}\\
 & \underline{64} \\
 & 8\\
\end{align} \,}} \right. \\
\end{align}\]

Step 3: Bring down the next pair. Here our next pair is $25$.
\[\begin{align}
  &\,\,\,\,\, 8 \\
 & 8\left| \!{\overline {\,
 \begin{align}
  & \overline{72}\overline{25}\\
 & \underline{64} \\
 & 825 \\
\end{align} \,}} \right. \\
\end{align}\]

Step 4: Now, double the value of the quotient and write it in divisor. For the second digit, we need to write such a number which when multiplied by the new number obtained gives us less than or equal to the dividend.
Here the quotient is $8$. So, the first part of the divisor will be $16$. Now, if we take the next part of the divisor as $5$, we get the product $165 \times 5 = 825$, equal to $825$.
So, we have,
\[\begin{align}
  & \,\,\,\,\,85 \\
 & 8\left| \!{\overline {\,
 \begin{align}
  & \overline{72}\overline{25}\\
 & \underline{64} \\
 & 825 \\
\end{align}
 \,}} \right. \\
 & 165\left| \!{\overline {\,
 \begin{align}
  & 825 \\
 & \underline{825} \\
 & 0\\
\end{align} \,}} \right. \\
\end{align}\]

Hence, we can conclude that $7225$ is the square of \[85\]. or the square root of $85$ is $7225$.

Note:
The given question could also be solved with a smart guess. The original number whose square root is to be calculated is $7225$. Now, we know that a square number having $5$ as the unit digit must have $5$ as the unit digit in the square root. Also, we know that the squares of numbers having $5$ as units digit can be calculated by prefixing the product of the first part of the number and the successor of that number in front of $25$.
So, we can observe that in $7225$, $72$ is the first part that is prefixed in front of $25$.
So, we have, $8 \times 9 = 72$.