# 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 n2. 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:

1. 121 – 1 = 120
2. 120 – 3 = 117
3. 117 – 5 = 112
4. 112 – 7 = 105
5. 105 – 9 = 96
6. 96 – 11 = 85
7. 85 – 13 = 72
8. 72 – 15 = 57
9. 57 – 17 = 40
10. 40 – 19 = 21
11. 21 – 21 = 0

Thus, we have subtracted consecutive odd numbers from 121 starting from 1. 0 is obtained in the 11th 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:

1. 81 – 1 = 80
2. 80 – 3 = 77
3. 77 – 5 = 72
4. 72 – 7 = 65
5. 65 – 9 = 56
6. 56 – 11 = 45
7. 45 – 13 = 32
8. 32 – 15 = 17
9. 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:

1. 49 – 1 = 48
2. 48 – 3 = 45
3. 45 – 5 = 40
4. 40 – 7 = 33
5. 33 – 9 = 24
6. 24 – 11 = 13
7. 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:

1. √625
2. √144
3. √64