Question

Find the square root of 156.25 using a long division method.
A. 12.3
B. 12.5
C. 12.4
D. 12.7

Answer 1

Hint:Mark the period in both integral and decimal parts in a pair. After that, find the square root by long division method and put the decimal point on the square root.

Complete step-by-step answer:
We have been asked to find the square root of 156.25. Let us consider the following steps to find the square root by long division method.
First draw lines over the pair of digits from right to left, i.e. \[1\overline{56.}\overline{25}\].
Now find the greatest number whose square is less than or equal to the digits in the first group. Now take this number as the divisor and quotient of the first group and find the remainder. Move the digits from the second group and find the remainder. Move the digits from the second group besides the remainder to get the new dividend. Now let us double the first divisor and bring it down as the new divisor. Now complete the divisor and continue the division.
Put the decimal point in the square root as soon as the integral part is exhausted. Now repeat the process till the remainder becomes zero.
So, 156.25 can be made into periods as \[1\overline{56.}\overline{25}\] from right to left.
Now with the following instruction find the square root of \[1\overline{56.}\overline{25}\].
\[1\overset{12.5}{\overline{\left){\begin{align}
  & 1\overline{56.}\overline{25} \\
 & \underline{1} \\
 & 22\overline{\left){\begin{align}
  & 056 \\
 & \underline{44} \\
 & 24.5\overline{\left){\begin{align}
  & 12.25 \\
 & \underline{12.25} \\
 & \underline{00.00} \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
\[\begin{align}
  & 21\times 1=21 \\
 & 22\times 2=44 \\
 & 23\times 3=69 \\
\end{align}\]
Similarly, \[\begin{align}
  & 24.5\times 5=12.25 \\
 & \therefore 24.5\times 0.5=12.25 \\
\end{align}\]
Thus we got remainder as zero.
Square root of 156.25 = 12.25.
\[\therefore \sqrt{156.25}=12.25\].
So option B is the correct answer.

Note:Remember that the division of the integral part must be by a perfect square, i.e. \[\left( 1\times 1 \right),\left( 2\times 2 \right),\left( 3\times 3 \right),\left( 4\times 4 \right)\] etc. which we have used here. Also remember to mark the periods in both integral and decimal parts on every pair of digits from left to right.