Question

Find the square root of 1331 using repeated subtraction.

Answer 1

Hint: To obtain the square root of the given number we will use repeated subtraction. Firstly we will subtract the successive odd numbers from the number whose square root is to be found till we get the answer as 0. Then we will count the steps used to get the value 0 which will be our square root and the desired answer.

Complete step by step answer:
We have to find the square root of:
 $1331$
We know that successive odd numbers are as follows:
$1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41........$
So starting from 1 till we get the answer as 0 we have the following steps:
$\begin{align}
  & 1331-1=1330 \\
 & 1330-3=1327 \\
 & 1327-5=1322 \\
 & 1322-9=1306 \\
\end{align}$
$\begin{align}
  & 1306-11=1295 \\
 & 1295-13=1282 \\
 & 1282-15=1267 \\
 & 1267-17=1250 \\
\end{align}$
$\begin{align}
  & 1250-19=1231 \\
 & 1231-21=1210 \\
 & 1210-23=1187 \\
 & 1187-25=1162 \\
\end{align}$
$\begin{align}
  & 1162-27=1135 \\
 & 1135-29=1106 \\
 & 1106-31=1075 \\
 & 1075-33=1042 \\
\end{align}$
$\begin{align}
  & 1042-35=1007 \\
 & 1007-37=970 \\
 & 970-39=931 \\
 & 931-41=890 \\
\end{align}$
$\begin{align}
  & 890-43=847 \\
 & 847-45=802 \\
 & 802-47=755 \\
 & 755-49=706 \\
\end{align}$
$\begin{align}
  & 706-51=655 \\
 & 655-53=602 \\
 & 602-55=547 \\
 & 547-57=490 \\
\end{align}$
$\begin{align}
  & 490-59=431 \\
 & 431-61=370 \\
 & 370-63=307 \\
 & 307-65=242 \\
\end{align}$
$\begin{align}
  & 242-67=175 \\
 & 175-69=106 \\
 & 106-71=35 \\
 & 35-73=-38 \\
\end{align}$
As we can observe that after repeated subtraction we are not getting our answer as 0 as after the ${{36}^{th}}$ step we are getting a negative value. So that means 1331 is not a perfect square.
So, the square root of 1331 lies between 36 and 37.

Hence the square root of 1331 is between 36 and 37.

Note: Before finding the square roots always check whether it is a perfect square or not because if it is not a perfect square using methods like prime factorization or repeated subtraction we will not get a perfect answer. The answer will be in the form of a range that the square root will be in. As we can see in this question as well we get that the answer is between 36 and 37 but we don’t know the exact value. So repeated subtraction method is not a good method when we are finding the square root of a non-perfect square.