Question

Find the square root of the following numbers using the long division method.
(a) 5929
(b) 2809
(c) 77841
(d) 12544
(e) 60516
(f) 33856
(g) 552049
(h) 119716

Answer 1

Hint: We will write the given number as a group of paired digits. Then we will use the long division method to find the square root of the number. We will consider the first pair of digits of the number and subtract from it the largest square which is less than the number formed by the first pair of digits. This is the first step of the long division method. Continuing with the next steps, we will obtain the square root.

Complete step-by-step answer:
(a) Let us look at the long division process in detail while computing the square root of 5929. The first step is to consider the first pair of digits from 5929 which is 59. We will subtract the largest square which is less than 59 from it. So, we have
$\begin{matrix}
   {} & 7 & {} & {} & {} \\
   7 & 5 & 9 & 2 & 9 \\
   - & 4 & 9 & \downarrow & \downarrow \\
   {} & 1 & 0 & 2 & 9 \\
\end{matrix}$
Now, we will double the quotient for the next step and add one more digit in the units place to it so that this new number multiplied to the digit in the units place to get the highest number which is less than the remainder. We will choose the digit 7, so that $147\times 7=1029$. Representing this in the long division format, we get the following,
$\begin{matrix}
   {} & 7 & 7 & {} & {} \\
   7 & 5 & 9 & 2 & 9 \\
   - & 4 & 9 & \downarrow & \downarrow \\
   147 & 1 & 0 & 2 & 9 \\
   - & 1 & 0 & 2 & 9 \\
   {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 5929 is 77.
For the next two numbers, we will directly write the long division form.
(b) 2809
$\begin{matrix}
   {} & 5 & 3 & {} & {} \\
   5 & 2 & 8 & 0 & 9 \\
   - & 2 & 5 & \downarrow & \downarrow \\
   103 & 0 & 3 & 0 & 9 \\
   - & {} & 3 & 0 & 9 \\
   {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 2809 is 53.
(c) 77841
Since this number has odd digits, we will exclude the first digit from forming a pair. So the long division will be as follows,
$\begin{matrix}
   {} & 2 & 7 & 9 & {} & {} \\
   2 & 7 & 7 & 8 & 4 & 1 \\
   - & 4 & \downarrow & \downarrow & {} & {} \\
   47 & 3 & 7 & 8 & {} & {} \\
   - & 3 & 2 & 9 & \downarrow & \downarrow \\
   549 & 0 & 4 & 9 & 4 & 1 \\
   - & {} & 4 & 9 & 4 & 1 \\
   {} & {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 77841 is 279.
(d) 12544
$\begin{matrix}
   {} & 1 & 1 & 2 & {} & {} \\
   1 & 1 & 2 & 5 & 4 & 4 \\
   - & 1 & \downarrow & \downarrow & {} & {} \\
   21 & 0 & 2 & 5 & {} & {} \\
   - & {} & 2 & 1 & \downarrow & \downarrow \\
   222 & {} & 0 & 4 & 4 & 4 \\
   - & {} & {} & 4 & 4 & 4 \\
   {} & {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 12544 is 112.
(e) 60516
$\begin{matrix}
   {} & 2 & 4 & 6 & {} & {} \\
   2 & 6 & 0 & 5 & 1 & 6 \\
   - & 4 & \downarrow & \downarrow & {} & {} \\
   44 & 2 & 0 & 5 & {} & {} \\
   - & 1 & 7 & 6 & \downarrow & \downarrow \\
   486 & {} & 2 & 9 & 1 & 6 \\
   - & {} & 2 & 9 & 1 & 6 \\
   {} & {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 60516 is 246.
(f) 33856
$\begin{matrix}
   {} & 1 & 8 & 4 & {} & {} \\
   1 & 3 & 3 & 8 & 5 & 6 \\
   - & 1 & \downarrow & \downarrow & {} & {} \\
   28 & 2 & 3 & 8 & {} & {} \\
   - & 2 & 2 & 4 & \downarrow & \downarrow \\
   364 & 0 & 1 & 4 & 5 & 6 \\
   - & {} & 1 & 4 & 5 & 6 \\
   {} & {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 33856 is 184.
(g) 552049
$\begin{matrix}
   {} & 7 & 4 & 3 & {} & {} & {} \\
   7 & 5 & 5 & 2 & 0 & 4 & 9 \\
   - & 4 & 9 & \downarrow & \downarrow & {} & {} \\
   144 & 0 & 6 & 2 & 0 & {} & {} \\
   - & {} & 5 & 7 & 6 & \downarrow & \downarrow \\
   1483 & {} & 0 & 4 & 4 & 4 & 9 \\
   - & {} & {} & 4 & 4 & 4 & 9 \\
   {} & {} & {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 552049 is 743.
(h) 119716
$\begin{matrix}
   {} & 3 & 4 & 6 & {} & {} & {} \\
   3 & 1 & 1 & 9 & 7 & 1 & 6 \\
   - & {} & 9 & \downarrow & \downarrow & {} & {} \\
   64 & 0 & 2 & 9 & 7 & {} & {} \\
   - & {} & 2 & 5 & 6 & \downarrow & \downarrow \\
   686 & {} & 0 & 4 & 1 & 1 & 6 \\
   - & {} & {} & 4 & 1 & 1 & 6 \\
   {} & {} & {} & {} & {} & {} & 0 \\
\end{matrix}$
Hence, the square root of 119716 is 346.

Note: The long division method is tricky. It is important to remember that we have to double the value of the quotient before adjoining a new digit in the units place. Otherwise we will get the wrong answer. Pairing of digits is crucial. For every step of the long division, we have to bring down the next pair of digits.