Question

What is the square root of $292.41$ using the long division method?
A) $17.1$
B) $17.2$
C) $17.3$
D) $17.4$

Answer 1

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 answer:
So, we will find the square root of the number $292.41$ 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.
\[\overline{2}\overline{92}.\overline{41}\]
Step 2:
For divisor, take the largest number whose square is less than or equal to the first pair of number. Here, first pair is $2$. So, $1$ is the largest number whose square is less than or equal to $2$. So, we will divide by $1$.
\[\begin{align}
  & 1 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{2}\overline{92}.\overline{41} \\
 & \underline{1} \\
 & 1 \\
\end{align} \,}} \right. \\
\end{align}\]
Step 3:
Bring down the next pair. Here our next pair is $92$.
\[\begin{align}
  & 1 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{2}\overline{92}.\overline{41} \\
 & \underline{1} \\
 & 192 \\
\end{align} \,}} \right. \\
\end{align}\]
Step 4:
Now, double the value of quotient and write it in divisor. For the second digit, we need to write such number which when multiplied by the new number obtained gives us less than or equal to dividend.
Here the quotient is $1$. So, one digit of the divisor will be $2$. Now, if we take the next part of divisor as $7$, we get the product $27 \times 7 = 189$, just less than $192$.
So, we have,
\[\begin{align}
  & 17 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{2}\overline{92}.\overline{41} \\
 & \underline{1} \\
 & 192 \\
\end{align}
 \,}} \right. \\
 & 27\left| \!{\overline {\,
 \begin{align}
  & 192 \\
 & \underline{189} \\
 & 3 \\
\end{align} \,}} \right. \\
\end{align}\]
Step 5: Now, taking the next pair down, we observe that there is a decimal point. So, we need to put a decimal point in the quotient as well and then take the next pair. Now, we get the first part of the divisor by doubling the quotient, that is $17 \times 2 = 34$.
Now, if we take the next part of divisor as $1$, we get the product $341 \times 1 = 341$.
So, we get,
\[\begin{align}
  & 17.1 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{2}\overline{92}.\overline{41} \\
 & \underline{1} \\
 & 192 \\
\end{align}
 \,}} \right. \\
 & 27\left| \!{\overline {\,
 \begin{align}
  & 192 \\
 & \underline{189} \\
 & 3 \\
\end{align} \,}} \right. \\
 & 341\left| \!{\overline {\,
 \begin{align}
  & 341 \\
 & \underline{341} \\
 & 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 $292.41$ is $17.1$.

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. The given question could also be solved with a smart guess. The original number is $292.41$ that means that the leftmost digit of the square root of the number must be either one or nine, since we know that numbers ending in $1$ or $9$ have the leftmost digit of square as $1$.
So, there is only one option $17.1$ with the leftmost digit as one or nine. Hence, option (A) is the correct answer.