# Square Root Of A Number By Repeated Subtraction

You have already learned about the squares and cubes of a number. 1, 4, 9, 16, 25, etc. are the squares of the numbers 1, 2, 3, 4, 5 and so on. In series 1, 4, 9…, the numbers are called** perfect squares** or square numbers. Thus, a square number can be defined as an integer that can be expressed as a product of a number with the number itself. And the number which is multiplied with itself is called the **square root** of the square number. So 25 is a square number that can be written as 5 X 5. And 5 is the square root of 25. Now finding the square of a number is simple. You multiply 10 with 10, and you obtain 100, which is the square of 10. But how do you go about finding the square root of a number? There are several methods for the same. In this article, we will learn how to find the square root of a number through repeated subtraction.

## Square Root by Repeated Subtraction

We know that the sum of the first *n* odd natural numbers is *n ^{2}*. We will use this fact to find the square root of a number by repeated subtraction. Let us take an example to learn this method. Say, you are required to find the square root of 121, that is, √121. The steps are:

- 121 – 1 = 120
- 120 – 3 = 117
- 117 – 5 = 112
- 112 – 7 = 105
- 105 – 9 = 96
- 96 – 11 = 85
- 85 – 13 = 72
- 72 – 15 = 57
- 57 – 17 = 40
- 40 – 19 = 21
- 21 – 21 = 0

Thus, we have subtracted consecutive odd numbers from 121 starting from 1. 0 is obtained in the 11^{th} step. So we have √121 = 11.

## Finding Square Root Through Repeated Subtraction

**Example 1:**

Find the square root of 81 using the repeated subtraction method.

**Solution:**

To find: √81

The steps to find the square root of 81 is:

- 81 – 1 = 80
- 80 – 3 = 77
- 77 – 5 = 72
- 72 – 7 = 65
- 65 – 9 = 56
- 56 – 11 = 45
- 45 – 13 = 32
- 32 – 15 = 17
- 17 – 17 = 0

Here, the result “0” is obtained in step 9. Hence, the square root of 81, √81 is 9.

**Example 2:**

Find the square root of 49 using the repeated subtraction method.

**Solution:**

To find: √49

The steps to find the square root of 49 is:

- 49 – 1 = 48
- 48 – 3 = 45
- 45 – 5 = 40
- 40 – 7 = 33
- 33 – 9 = 24
- 24 – 11 = 13
- 13 – 13 = 0

The result “0” is obtained in the 7th step.

Hence, the square root of 49, √49 is 7.

### Practice Problems

Find the square root for the given numbers using repeated subtraction:

- √625
- √144
- √64

Click on the linked article to learn more about square roots of decimals, and know more at byjus.com