Question

Find the square root of each of the following by long division method:
120409

Answer 1

Hint: Here, we have to find the square root of 120409. Square root of a number means the number which when multiplied by itself gives the original number. The square root of a number is denoted by the radical sign $\sqrt {} $ and the value inside this is known as radicand. Here, we are going to use the long division method to find the square root of 120409.

Complete step-by-step answer:
In this question, we are given a number and we have to find its square root using a long division method.
Given number: 120409
Now, we can find the square root of a number in many ways but in this question, we are asked to use the long division method so we will use the long division method only.
A square root of a number means the number when multiplied by itself gives the original number. Let us see the steps to find the square root using the long division method.
Steps to find square root using Square root:
Step 1:
Write down the number whose square root is to be found and place a bar over the pairs of two numbers from units place.
$\overline{12}\overline{04}\overline{09}$
Step 2:
For the divisor, take the largest number whose square is less than or equal to the first pair of numbers. Here, the first pair is 12. So, 3 is the largest number whose square is less than 12. So, we will divide by 3.
\[\begin{align}
  &\;\;\;\;\;\; 3 \\
 & 3\left| \!{\overline {\,
 \begin{align}
  & \overline{12}\overline{04}\overline{09} \\
 & \underline{09} \\
 & 03 \\
\end{align} \,}} \right. \\
\end{align}\]
Step 3:
Bring down the next pair. Here our next pair is 04.
\[\begin{align}
  &\;\;\;\;\;\;\;\; 3 \\
 & 3\left| \!{\overline {\,
 \begin{align}
  & \overline{12}\overline{04}\overline{09} \\
 & \underline{09} \\
 & 0304 \\
\end{align} \,}} \right. \\
\end{align}\]
Step 4:
Now, double the value of the quotient and write it in the divisor. For the second digit, we need to write such a number which when multiplied by the new number obtained gives us less than or equal to the dividend.
Here the quotient is 3. So, one digit of the divisor will be 6. Now, if we take the second digit 4 and then multiply 64 by 4, we get the product less than our dividend.
$\Rightarrow 64\times 4=256$
\[\begin{align}
  & \;\;\;\;\;\; 34 \\
 & 3\left| \!{\overline {\,
 \begin{align}
  & \overline{12}\overline{04}\overline{09} \\
 & \underline{09} \\
 & 03 \\
\end{align} \,}} \right. \\
 & 64\left| \!{\overline {\,
 \begin{align}
  & 304\overline{09} \\
 & \underline{256} \\
 & 048 \\
\end{align} \,}} \right. \\
\end{align}\]
Step 5:
Now, bring down the third pair. Here, our third pair is 09.
\[\begin{align}
  &\;\;\;\;\;\;34 \\
 & 64\left| \!{\overline {\,
 \begin{align}
  & 304\overline{09} \\
 & \underline{256} \\
 & 04809 \\
\end{align} \,}} \right. \\
\end{align}\]
Step 6:
Now, add the last quotient to our last divisor. So our last quotient was 4 and our last divisor was 64. Therefore, the next divisor will be
$\Rightarrow 64+4=68\_$
Now, for the third digit go with step 4.
So, here if we take the third digit equal to 7 and then multiply 687 with 7, we get an answer equal to the dividend.
$\Rightarrow 687\times 7=4809$
\[\begin{align}
  &\;\;\;\;\;\;\; 347 \\
 & 687\left| \!{\overline {\,
 \begin{align}
  & 4809 \\
 & \underline{4809} \\
 & 0 \\
\end{align} \,}} \right. \\
\end{align}\]
Hence, the square root of 120409 is 347.

Note: If this method seems complicated to you, you can also use the method of prime factorization to find the square root of 120409.
$ \Rightarrow \sqrt {120409} = \sqrt {347 \times 347} $
Here, we can see that 347 is repeated 2 times.
$ \Rightarrow \sqrt {120409} = 347$
Hence, the square root of 120409 is 347.