What is the square root of $21.16$ using the long division method?

Hint: In the given question, we need to find the square root of the given number using the long division method. Square root of a number is a number that when multiplied with itself yields the original number. 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 $21.16$ 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.
\[21.16\]

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 $21$. So, $4$ is the largest number whose square is less than or equal to $21$. So, we will divide by $4$.
\[\begin{align}
  & \,\,\,\,4 \\
 & 4\left| \!{\overline {\,
 \begin{align}
  & \overline{21}.\overline{16} \\
 & \underline{16} \\
 & 5 \\
\end{align} \,}} \right. \\
\end{align}\]

Step 3: Bring down the next pair. Here our next pair is $16$.
\[\begin{align}
  &\,\,\,\, 4 \\
 & 4\left| \!{\overline {\,
 \begin{align}
  & \overline{21}.\overline{16} \\
 & \underline{16} \\
 & 516 \\
\end{align} \,}} \right. \\
\end{align}\]

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 $4$. So, the first part of the divisor will be $8$. Now, if we take the next part of the divisor as $6$, we get the product $86 \times 6 = 516$. So, we have,
\[\begin{align}
  &\,\,\,\, 4.6 \\
 & 4\left| \!{\overline {\,
 \begin{align}
  & \overline{21}.\overline{16} \\
 & \underline{16} \\
 & 516 \\
\end{align}
 \,}} \right. \\
 & 86\left| \!{\overline {\,
 \begin{align}
  & 516 \\
 & \underline{516} \\
 & 0 \\
\end{align} \,}} \right. \\
\end{align}\]
Now, the quotient obtained in the long division process is the square root of the given number.
Hence, the square root of $21.16$ is $4.6$. Thus, option (A) is the correct answer.

Note:
We must remember to place the decimal point once we arrive at the position of the decimal point in the dividend or original number.
We can also cross check the final answer by calculating the square of the number $4.6$ and checking if it is equal to $21.16$ or not.
So, we get,
${\left( {4.6} \right)^2} = 4.6 \times 4.6 = 21.16$
Hence, the final answer is correct and $4.6$ is the square root of $21.16$.