Question

What is the square root of \[337\]?

Answer 1

Hint: First we will check if we can use the prime factorization method to find the square root or not. If we will be able to write all the prime factors as exponent equal to 2 then we will be able to find the root easily. If this method fails then we will apply the long division method to get our answer in decimal up to two places.

Complete step by step answer:
Here we have been asked to find the square root of 337.
Now, 337 is a prime number so we will not be able to write the factors with their exponent equal to 2. Therefore, we need to apply the long division process to find the square root. So, let us see each step while finding the root.
Step (1): Starting from the rightmost digit we will form pairs of two digits till all the digits are paired or a single digit is left at the leftmost place, here we have three digits so 37 will be paired and 3 will be left.
Step (2): We will select a number whose square will be less than or equal to 3. Here we have to select 1. Now, we will write 1 as the divisor and the quotient and subtract the product from 3. The remainder will be 2. Now, we will write the initially formed pair 37 beside this remainder.
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,1 \\
 & \begin{matrix}
   \,\,\,\,\,\,\,1 \\
   \,\,\,+1 \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \,\,\,\,\,\,\,3\overline{37} \\
 & \,\,\,-1 \\
 & \overline{\,\,\,\,\,\,\,2\,37} \\
\end{align} \,}} \right. \\
\end{align}\]
Step (3): The new dividend thus becomes 237, at the divisor place we will get the sum 1 + 1 = 2. Now, we need to select a digit after 2 such that when we will take the product of this selected digit with the new number we will get a number less than 237. So here we will select 8. The new divisor will become 28 + 8 = 36, the remainder will be 13 and the new quotient will become 18.
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,18 \\
 & \begin{matrix}
   \,\,\,\,\,\,\,\,1 \\
   \,\,\,\,+1 \\
   \overline{\,\,\,\,\,\,\,\,\,28} \\
   \,\,\,\,\,\,\,+8 \\
   \overline{\,\,\,\,\,\,\,\,36} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \,\,\,\,\,\,\,3\overline{37} \\
 & \,\,\,-1 \\
 & \overline{\,\,\,\,\,\,\,\,\,237} \\
 & \,\,\,\,\,-224 \\
 & \overline{\,\,\,\,\,\,\,\,\,\,\,\,13} \\
\end{align} \,}} \right. \\
\end{align}\]
Step (4): Now, there is no more paired digits to be placed besides the remainder 12 so we will take the decimal point in the quotient and place two 0’s after 13. The dividend thus becomes 1300. Further we need to select a digit after 36 such that the product of this digit with the new number will be less than 1300, so we will select 3.
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,18.3 \\
 & \begin{matrix}
   \,\,\,\,\,\,\,1 \\
   \,\,\,+1 \\
   \overline{\,\,\,\,\,\,\,\,\,28} \\
   \,\,\,\,\,\,\,+8 \\
   \overline{\,\,\,\,\,\,\,\,\,\,\,363} \\
   \,\,\,\,\,\,\,\,\,\,\,\,+3 \\
   \overline{\,\,\,\,\,\,\,\,\,\,\,366} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \,\,\,\,\,\,\,337 \\
 & \,\,\,-1 \\
 & \overline{\,\,\,\,\,\,\,237} \\
 & \,\,\,-224 \\
 & \overline{\,\,\,\,\,\,\,\,\,\,1300} \\
 & \,\,\,\,\,\,-1089 \\
 & \overline{\,\,\,\,\,\,\,\,\,\,\,\,\,211} \\
\end{align} \,}} \right. \\
\end{align}\]
Step (5): Now, we will get the divisor 363 + 3 = 336 and the remainder will be 211. The quotient is now 18.3. Again two 0’s will be placed after the remainder so that it becomes 21100 due to the decimal point. Now, we need to select 5 beside the divisor so that the product of 3365 and 5 will be less than 21100. Therefore, the quotient will become 18.35.
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,18.35 \\
 & \begin{matrix}
   \,\,\,\,\,\,1 \\
   \,\,+1 \\
   \overline{\,\,\,\,\,\,\,\,\,28} \\
   \,\,\,\,\,\,\,+8 \\
   \overline{\,\,\,\,\,\,\,\,\,\,\,363} \\
   \,\,\,\,\,\,\,\,\,\,\,\,+3 \\
   \,\,\,\,\,\,\,\overline{\,\,\,\,\,\,\,3665} \\
   \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+5 \\
   \,\,\,\,\,\,\,\,\,\overline{\,\,\,\,\,3670} \\
\end{matrix}\left| \!{\overline {\,
 \begin{align}
  & \,\,\,\,\,\,\,3\overline{37} \\
 & \,\,\,-1 \\
 & \overline{\,\,\,\,\,\,\,237} \\
 & \,\,\,-224 \\
 & \overline{\,\,\,\,\,\,\,\,\,\,1300} \\
 & \,\,\,\,\,\,-1089 \\
 & \overline{\,\,\,\,\,\,\,\,\,\,\,\,21100} \\
 & \,\,\,\,\,\,\,\,-18325 \\
 & \overline{\,\,\,\,\,\,\,\,\,\,\,\,\,\,2775} \\
\end{align} \,}} \right. \\
\end{align}\]
Hence, the square root of 337 is 18.35 correct up to two places of decimal.

Note: You must remember the long division method to find the square root because this method is very useful in determining the square roots of numbers which are not perfect squares. You can proceed further in the similar manner to find more digits after the point we have stopped above. Generally, we do not go beyond three places of decimal in our answer.