Question

Find the square root of 144 by the method of repeated subtraction.

Answer 1

Hint: Now, as we know that series of odd numbers is $1,3,5,7.............\left( 2n-1 \right)$ . Where we can vary the value of n and we will get the next term of the series.

Complete step by step answer:
The sum of the first $n$ odd numbers $=1+3+5+7+............+\left( 2n-1 \right)=\sum\limits_{r=1}^{n}{\left( 2r-1 \right)={{n}^{2}}}$ .

We can use the above formula to find the square root of any perfect square by using the following steps:

1. Firstly write the number whose square root is to be found, then start subtracting odd numbers from the number starting from one.
2. Keep subtracting odd numbers till we get zero.
3. When we get zero then, find the last odd number we subtracted and equate it to $\left( 2n-1 \right)$ and find the value of $n$ , then $n$ will be the square root of that number.

Complete step-by-step answer:

We have to find the square root of 144.

We can use the above-mentioned steps to find the square root of any perfect square. This method is known as the method of repeated subtraction. It is very easy to understand and use.

Now, we find the square root of 144 using the method of repeated subtraction. Subtracting odd numbers from 144 till we get 0. Then,
$\begin{align}
  & 144-1=143 \\
 & \Rightarrow 143-3=140 \\
 & \Rightarrow 140-5=135 \\
 & \Rightarrow 135-7=128 \\
 & \Rightarrow 128-9=119 \\
 & \Rightarrow 119-11=108 \\
 & \Rightarrow 108-13=95 \\
 & \Rightarrow 95-15=80 \\
 & \Rightarrow 80-17=63 \\
 & \Rightarrow 63-19=44 \\
 & \Rightarrow 44-21=23 \\
 & \Rightarrow 23-23=0 \\
\end{align}$

Now, as the last odd number that we subtracted is $23$ . So, now we will find the value of $n$ from the equation: $\left( 2n-1 \right)=23$ . Then,
$\begin{align}
  & \left( 2n-1 \right)=23 \\
 & \Rightarrow 2n=24 \\
 & \Rightarrow n=12 \\
\end{align}$

Now, as the value of $n=12$ . So, the square root of 144 will be 12.

Thus, the square root of 144 is 12.

Note: The question is very easy to simple if we know what method of repeated subtraction and we can use it to find the square root of any perfect square. Moreover, we should perform subtraction carefully without any calculation mistakes to get the correct answer.