Question

Find the square root of 625 by long division method.

Answer 1

Hint: To find out the square root of 625 by long division method we will start with putting bars on two digits and starting from the unit place of the number. We will take the large number as the divisor whose square is less than or equal to the dividend. We will write the quotient which will be the same as the divisor. We will subtract the square from the dividend. We will get a number. Then we will bring the next pair down. This will become the new dividend. The first digit of the divisor will be the double of the quotient and the next digit will be chosen such that the product of the divisor and the digit chosen is just less than the dividend. The digit chosen will be the next digit of the quotient. We will repeat the steps until we get a zero. The quotient will be the square root of the required number.

Complete step-by-step answer:
To start with, we will write the digits of 625 in pairs starting from the unit digit by writing the bars over them. Thus, we have \[\overline{6}\text{ }\overline{25}.\] Now, we will select the leftmost digit, i.e. 6 as the dividend. Now, we will select the quotient and the divisor such that their square is just less than 6. We know that, \[{{2}^{2}}=4\] and \[{{3}^{2}}=9.\] Thus, the quotient and the divisor will be 2. Now, we will subtract the square of the quotient from the divisor. Thus, we will get,
\[2\overset{2}{\overline{\left){\begin{align}
  & \begin{matrix}
   \overline{6} & \overline{25} \\
\end{matrix} \\
 & \underline{-4} \\
 & 2 \\
\end{align}}\right.}}\]

Now, we will bring the next pair of digits down to the right to the remainder. The obtained number will be the new dividend. Thus, we have,
\[2\overset{2}{\overline{\left){\begin{align}
  & \begin{matrix}
   \overline{6} & \overline{25} \\
\end{matrix} \\
 & \underline{-4} \\
 & \begin{matrix}
   2 & \overline{25} \\
\end{matrix} \\
\end{align}}\right.}}\]

Thus, the new dividend is 225. Now, the first digit of the divisor will be the double of the quotient, i.e. \[2\times 2=4.\]The next digit will be chosen such that when the divisor is multiplied with that digit, the number obtained is just less than or equal to the dividend. Thus, the divisor is of the form of 4N. Now, if we choose N as 5, we will get \[45\times 5=225\] which is just equal to 225. Thus, the divisor in this case will be 45. The next digit of the quotient will be the digit is chosen, i.e. 5. Thus, we have,
\[\] \[\begin{align}
  & 2\overset{25}{\overline{\left){\begin{align}
  & \begin{matrix}
   \overline{6} & \overline{25} \\
\end{matrix} \\
 & \underline{-4} \\
 & \begin{matrix}
   2 & \overline{25} \\
\end{matrix} \\
\end{align}}\right.}} \\
 & 45\overline{\left){\begin{align}
  & \begin{matrix}
   2 & \overline{25} \\
\end{matrix} \\
 & \underline{\begin{matrix}
   2 & \overline{25} \\
\end{matrix}} \\
 & 0 \\
\end{align}}\right.} \\
\end{align}\]

Now, we have obtained the remainder as 0. Thus, the process ends now. The value of the quotient will be the square root of 625. The quotient is 25. Thus, we have,
Square root of 625 = 25

Note: It is not always necessary that we zero as the remainder. We may have the remainder of the divisor as non-zero in every step. This means that the number whose square root we are calculating is not a perfect square of an integer. We will get zero in the remainder if the number is the perfect square of the integer.