Question

Find the square root of 531.7636
(a) 23.06
(b) 23.03
(c) 23.01
(d) 23.10

Answer 1

Hint: To solve the given question, we will use the division method to find the square root of 531.7636. In this method, first, we will group the digits in pairs, starting with the digits in the unit place. Then we will follow the certain steps of the division method. We will take the digit(s) in the first place as dividends and then we will find the square of the number which is less than or equal to the dividend. We will take both dividend and quotient as that number. Now, we will subtract the square from the divisor and we will add the next pair on this number. We will now double the quotient and take it as the next divisor and add another integer next to it such that the product of the number obtained and the integer added is less than the new dividend and we will subtract this product from it. We will continue doing this till we get the result.

Complete step by step answer:
Here, to find the square root of 531.7636, we will use the division method. In this method, first, we will group the digits in pairs with the help of the bar above them. The grouping will start from the unit place. Thus, we have,
$\left. {} \right)\overline{5}\text{ }\overline{31}.\text{ }\overline{76}\text{ }\overline{36}\left( {} \right.$.
Now we will find the quotient and the divisor. In the starting, we will take 5 as dividend. The divisor will be that number whose square is equal to or less than the dividend. We can say that, \[{{2}^{2}}=4\] (less than 5) so the divisor will be 2. The quotient will also be equal to the divisor i.e. 2. Now, we will subtract the square of 2 from the dividend. Thus, we will get,
$\begin{align}
  & \left. 2 \right)\overline{5}\text{ }\overline{31}.\text{ }\overline{76}\text{ }\overline{36}\left( 2 \right. \\
 & \underline{\text{ 4 }} \\
 & \text{ 1} \\
\end{align}$.
Now, we will bring down the next pair which is 31. Thus, we will get,
$\begin{align}
  & \left. 2 \right)\overline{5}\text{ }\overline{31}.\text{ }\overline{76}\text{ }\overline{36}\left( 2 \right. \\
 & \underline{\text{ 4 }} \\
 & \text{ 1}31 \\
\end{align}$.
Now, 131 will be the new dividend. To find the new divisor, we will double the quotient and add a one-digit number next to it such that when the divisor is multiplied by that unit digit number, it will be less or equal to the dividend. If the divisor is of the form 4x then, \[\left( 4x \right).\left( x \right)\le 131.\] Here, when we take x = 3, we get,
\[43\times 3\le 131\]
\[\Rightarrow 129\le 131\]
Thus, the value of x = 3. 3 will be the next digit in the quotient. Now, we will subtract the product from 131. Thus, we will get,
$\begin{align}
  & \text{ }\left. 2 \right)\overline{5}\text{ }\overline{31}.\text{ }\overline{76}\text{ }\overline{36}\left( 23 \right. \\
 & \underline{\text{ 4 }} \\
 & \left. 43 \right)\text{1}31 \\
 & \underline{\text{ 129 }} \\
 & \text{ 2} \\
\end{align}$.
Now, similarly, we will bring down the next pair. Then we will get,
$\begin{align}
  & \text{ }\left. 2 \right)\overline{5}\text{ }\overline{31}.\text{ }\overline{76}\text{ }\overline{36}\left( 23 \right. \\
 & \underline{\text{ 4 }} \\
 & \left. 43 \right)\text{1}31 \\
 & \underline{\text{ 129 }} \\
 & \text{ 2}.76 \\
\end{align}$.
Now, 276 is the new dividend. Now we will repeat the same steps. This time, we will get a divisor as 460 and the quotient will be 0. Thus we will get,
$\begin{align}
  & \text{ }\left. 2 \right)\overline{5}\text{ }\overline{31}.\text{ }\overline{76}\text{ }\overline{36}\left( 23.0 \right. \\
 & \underline{\text{ 4 }} \\
 & \text{ }\left. 43 \right)\text{1}31 \\
 & \underline{\text{ 129 }} \\
 & \left. 460 \right)\text{2}.76 \\
 & \underline{\text{ 0 }} \\
 & \text{ 2}\text{.76} \\
\end{align}$.
Now, we will bring down the next pair of numbers, i.e. 36. Now, we will take 27636 as the next dividend. The next divisor will be 4606 and the next quotient will be 6. Thus, we will get,
$\begin{align}
  & \text{ }\left. 2 \right)\overline{5}\text{ }\overline{31}.\text{ }\overline{76}\text{ }\overline{36}\left( 23.06 \right. \\
 & \underline{\text{ 4 }} \\
 & \text{ }\left. 43 \right)\text{1}31 \\
 & \underline{\text{ 129 }} \\
 & \left. \text{ }460 \right)\text{2}76 \\
 & \underline{\text{ 0 }} \\
 & \left. 4606 \right)\text{276}46 \\
 & \underline{\text{ 27646 }} \\
 & \text{ 0} \\
\end{align}$.
Now, the remainder is obtained as zero, so the divisor terminates and the quotient will be equal to the square root of the 531.7646. Thus, we have,
The square root of 531.7646 = 23.06
Hence, option (a) is the right answer.

Note:
The square root of 531.7636 can also be obtained by the alternate method shown.
We can write 531.7636 as \[5317636\times {{10}^{-4}}.\] Now, we will first find the square root of 5317636 using the prime factorization method. Thus, we will take the LCM of 5317636.
$\begin{align}
  & \text{ }2\left| \!{\underline {\,
  5317636 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  2658818 \,}} \right. \\
 & 1153\left| \!{\underline {\,
  1329409 \,}} \right. \\
 & 1153\left| \!{\underline {\,
  1153 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$.
Thus,
\[5317636=2\times 2\times 1153\times 1153\].
\[\Rightarrow 5317636={{2}^{2}}\times {{1153}^{2}}\].
Thus, we can say that,
\[\Rightarrow 531.7636={{2}^{2}}\times {{1153}^{2}}\times {{10}^{-4}}\].
Taking square root on both the sides, we get,
\[\Rightarrow \sqrt{531.7636}=\sqrt{{{2}^{2}}\times {{1153}^{2}}\times {{10}^{-4}}}\].
\[\Rightarrow \sqrt{531.7636}=2\times 1153\times {{10}^{-2}}\].
\[\Rightarrow \sqrt{531.7636}=23.06\].