Question

Find the square root of 4489 using repeated subtraction.

Answer 1

- Hint:- To get a square root of 4489, keep subtracting odd numbers from the odd consecutive natural number. Keep subtracting until we get zero. Thus the last step indicates the square root of 4489.

Complete step-by-step answer: -
We need to find the square root of 4489 by using repeated subtraction. The square of any natural number n can be represented as the sum of n consecutive odd numbers. So to get the square root of 4849, we need to keep subtracting odd natural numbers until we get zero.
Step 1: 4489 – 1 = 4488
Step 2: 4489 – 3 = 4485
Step 3: 4485 – 5 = 4480
Step 4: 4480 – 7 = 4473
Step 5: 4473 – 9 = 4464
Step 6: 4464 – 11 = 4453
.
.
.
Step 67: 133 – 133 = 0
Thus by subtracting consecutive odd numbers in the same manner, we get the remainder as zero.
Thus 4489 is obtained in the 67th step. This is the square root of 4485 is 67.
\[\therefore \sqrt{4485}=67.\]

Note:
The method of repeated subtraction is possible only for perfect square numbers. If the number is not a perfect square, then the remainder will never be zero. We can also find the square root by long division method.
\[6\overset{67}{\overline{\left){\begin{align}
  & \overline{44}\overline{89} \\
 & \underline{36} \\
 & 127\overline{\left){\begin{align}
  & 889 \\
 & \underline{889} \\
 & \underline{000} \\
\end{align}}\right.} \\
\end{align}}\right.}}\] i.e. \[\sqrt{4489}=67\]