By using the table for square root, find the value of \[13.21\] and \[21.97\]?
(A) \[3.63,4.60\]
(B) \[3.63,4.69\]
(C) \[3.53,4.69\]
(D) \[3.63,4.19\]

Hint: We are given two numbers whose square roots have to be found. We will use the long division method to find the same. For \[13.21\], we will first pair up the digits from right to left for the digits left to the decimal and left to right for the digits right to the decimal and then we will divide the pairs by the nearest square. Consequently, we will have the square root of the \[13.21\] in the quotient. Similarly, we will search for \[21.97\] and find its square root.

Complete step by step solution:
According to the given question, we are given two numbers whose square roots we have to find.
Square root of a number can be said to be a value which when multiplied by itself gives the number.
For example – we know that the square root of 25 is 5, that is, \[\sqrt{25}=5\] and that also means that the square of 5 will give us 25, that is, \[{{5}^{2}}=25\].
The first number we have is, \[13.21\].
We will be using the long division method to find the square root of \[13.21\].
We have,
\[\overline{\left){13.21}\right.}\]
So, firstly, we will pair up the digits starting from the decimal to either of the sides. That is,
\[\overline{\left){\underline{13}.\underline{21}}\right.}\]
But division will remain the same as before, we will go from left to right.
Now , to divide the first pair which is \[13\], we need a square nearest to \[13\]. We will now look at the table of squares for numbers 1 to 9, we have,

 We can see that, 13 lies between 9 and 16, that is,
\[9 < 13 < 16\]
\[\Rightarrow {{3}^{2}} < 13 < {{4}^{2}}\]
So, we will divide by 3 first, we have,
\[3\overset{3}{\overline{\left){\begin{align}
  & 13.21 \\
 & \underline{-9} \\
 & 4 \\
\end{align}}\right.}}\]
We got the remainder as 4, and now we will take the next pair, so we have,
\[3\overset{3}{\overline{\left){\begin{align}
  & 13.21 \\
 & \underline{-9} \\
 & 4.21 \\
\end{align}}\right.}}\]
Now, we will be dividing the remainder by twice the quotient which is 6, we get,
\[3\overset{3}{\overline{\left){\begin{align}
  & 13.21 \\
 & \underline{-9} \\
 & 6\_\overline{\left){4.21}\right.} \\
\end{align}}\right.}}\]
We need a unit place digit for the divisor such that resulting product is a number nearest and lower to \[4.21\] and that same unit place digit will be written in the unit place of the quotient, we have,
x\[3\overset{3.6}{\overline{\left){\begin{align}
  & 13.21 \\
 & \underline{-9} \\
 & 66\overline{\left){\begin{align}
  & 4.21 \\
 & \underline{-3.96} \\
 & 0.25 \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
We will find the square root till two decimal places, next we will have the dividend with a pair of zeroes and the divisor will be twice the quotient with a suitable unit place digit and on dividing, we get,
\[3\overset{3.63}{\overline{\left){\begin{align}
  & 13.21 \\
 & \underline{-9} \\
 & 66\overline{\left){\begin{align}
  & 4.21 \\
 & \underline{-3.96} \\
 & 723\overline{\left){\begin{align}
  & 0.2500 \\
 & \underline{-0.2169} \\
 & 0.0331 \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
Therefore, \[\sqrt{13.21}=3.63\]
Similarly, the square root of \[21.97\]is also found using the above steps.
 We got,
We will first pair the digits,
\[\overline{\left){\underline{21}.\underline{97}}\right.}\]
Then, we will find a square nearest and lower to the first pair and divide the dividend by the number of that square. And then we will proceed in the same way as we did for the above question. We get,
\[4\overset{4.68}{\overline{\left){\begin{align}
  & 21.97 \\
 & \underline{-16} \\
 & 86\overline{\left){\begin{align}
  & 5.97 \\
 & \underline{-5.16} \\
 & 928\overline{\left){\begin{align}
  & 0.8100 \\
 & \underline{-0.7424} \\
 & 0.0676 \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
Since, in the options there is no such answer so we will find one more decimal unit. We have,
\[4\overset{4.687}{\overline{\left){\begin{align}
  & 21.97 \\
 & \underline{-16} \\
 & 86\overline{\left){\begin{align}
  & 5.97 \\
 & \underline{-5.16} \\
 & 928\overline{\left){\begin{align}
  & 0.8100 \\
 & \underline{-0.7424} \\
 & 9367\overline{\left){\begin{align}
  & 0.067600 \\
 & \underline{-(0.065569)} \\
 & \underline{0.002031} \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
Therefore, \[\sqrt{21.97}=4.687\approx 4.69\]

So, the correct answer is “Option B”.

Note: The long division method is to be done carefully and values should be misread or calculated wrongly. Also, the square of the number should be written correctly and for that table of the squares must be referred. While substituting the values in the long division method, the divisor is the number and not the number’s square.