Question

Find the square root of the following numbers using division method: 651.7809

Answer 1

Hint: In order to solve this problem, we need to make pairs of two numbers starting from either side of the decimal point. Then we need to take one pair at a and perform the division method. We need to add the numbers in the left column and subtract the product from the right column.

Complete step-by-step answer:
We need to find the square root of this decimal number.
To do this first we need to make the pairs of two numbers starting from either side of the decimal point.
It is shown as follows,
 $ \overline{6}\overline{51}.\overline{78}\overline{09}..............(i) $
Now we will use the division method to find the solution.
We need to consider the numbers according to their respective pairs.
Therefore, we get,


Therefore, we get,

	2
\[\begin{align} & \,\,\,2 \\ & +2 \\ \end{align}\]	$\begin{align} & \overline{\,\,\,\,6}\overline{51}.\overline{78}\overline{09} \\ & -4 \\ \end{align}$
4	2

Therefore, we can see that the square just near to 6 is 4.
Now, we need to take the next pair,

	25
\[\begin{align} & \,\,\,2 \\ & +2 \\ \end{align}\]	$\begin{align} & \overline{\,\,\,\,6}\overline{51}.\overline{78}\overline{09} \\ & -4 \\ \end{align}$
$\begin{align} & 45 \\ & +5 \\ \end{align}$	$\begin{align} & \,\,\,251 \\ & -225 \\ \end{align}$
50	$\,\,\,26$

We can see that in the next step we have to find the multiplication such that the unit’s place is the same.
45 x 5 = 225 < 251, therefore, we can use this as 46 x 6 >251.
The same number is added on the left column.
Following the same process for the rest of the process.
We now need to put the decimal point after 25 as in the main number the decimal point starts.

	25.5
\[\begin{align} & \,\,\,2 \\ & +2 \\ \end{align}\]	$\begin{align} & \overline{\,\,\,\,6}\overline{51}.\overline{78}\overline{09} \\ & -4 \\ \end{align}$
$\begin{align} & 45 \\ & +5 \\ \end{align}$	$\begin{align} & \,\,\,251 \\ & -225 \\ \end{align}$
$\begin{align} & 505 \\ & +\,\,5 \\ \end{align}$	$\begin{align} & \,\,\,2678 \\ & -2525 \\ \end{align}$
510	$\,\,\,153$

We need to repeat the same process after the decimal point. We can see that 505 x 5 = 2525 < 2678.
By doing it one more time we get,

	25.53
\[\begin{align} & \,\,\,2 \\ & +2 \\ \end{align}\]	$\begin{align} & \overline{\,\,\,\,6}\overline{51}.\overline{78}\overline{09} \\ & -4 \\ \end{align}$
$\begin{align} & 45 \\ & +5 \\ \end{align}$	$\begin{align} & \,\,\,251 \\ & -225 \\ \end{align}$
$\begin{align} & 505 \\ & +\,\,5 \\ \end{align}$	$\begin{align} & \,\,\,2678 \\ & -2525 \\ \end{align}$
$5103$	$\begin{align} & \,\,\,15309 \\ & -15309 \\ \end{align}$
	$\,\,\,\,0$

We can see that we did get the remainder a zero at the end of the process.
Hence the square root of 651.7809 is 25.53.

Note: We need to subtract the product and we need to add the numbers on the left column.Also, we need to continue the process until the remainder zero comes. We need to continue this by adding two zeros at a time because zeros do not change the value after the decimal point.