Square Root by Repeated Subtraction
We have already learnt the square root and cube root of a number. The square root of a number, is that number which when multiplied by itself gives the number itself. That number is called a perfect square. For Example,the square root of 4 is 2, the square root of 9 is 3 and the square root of 16 is 4 and so on. Square root of a number is denoted by the symbol √ .Just as the division is the inverse operation of multiplication, the square root is the inverse operation of squaring a number. So to find out the square root of a number is opposite to finding the square of a number. For example square of 4 is 16 and square root of 16 is 4. Squaring of a number is an easy task but then how to find the square root of a number. Finding the square root of a number by repeatedly subtracting successive odd numbers from the given square number, till you get zero is known as repeated subtraction method.
Finding Square Roots:
First check whether the given number is a perfect square number or not. If it is a product of any number by itself then it is a perfect square To find the square root of perfect square numbers, any one of the following methods can be used.

Prime factorization method

Repeated subtraction method

Long division method

Number line method

Average method
But, if the number is not a perfect square prime factorization method and the repeated subtraction method will not work, we have to use other methods for finding the square roots.
In this article let us study how to find the square root by repeated subtraction method.
Square Root by Repeated Subtraction
We know that by the property of square numbers, if a natural number is a square number, then it has to be the sum of successive odd numbers starting from 1. We will be using this property to find out the square root of a number by repeated subtraction method.Therefore, you can find the square root of a number by repeatedly subtracting successive odd numbers from the given square number, till you get zero.
Let us study how to find the square root of 121 by repeated subtraction method.
Square Root of 121 by Repeated Subtraction
To find the square root of \[\sqrt{121}\] by repeated subtraction we will subtract successive odd numbers starting from 1 from 121. The number of steps obtained to get the result 0 is the square root of the given number. To find square root of 121 follow the steps given below :
Step1 : 121 – 1 = 120
Step 2: 120 – 3 = 117
Step 3: 117 – 5 = 112
Step 4: 112 – 7 = 105
Step 5: 105 – 9 = 96
Step 6: 96 – 11 = 85
Step 7: 85 – 13 = 72
Step 8: 72 – 15 = 57
Step 9: 57 – 17 = 40
Step 10: 40 – 19 = 21
Step 11: 21 – 21 = 0
Here, we got the result 0 in the 11th step, so we can say that \[\sqrt{121}\] = 11.
Let us consider another example to find the square root of 81 by repeated subtraction.
Square Root of 81 by Repeated Subtraction
Let us find the square root of 81 by repeated subtraction method.

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 we got the result 0 in the 9th step, so the square root of 81 is 9 i.e \[\sqrt{81}\] = 9
The Square roots of 1 to 25 are listed in the table below.
Square Root 1 to 30
Fun Facts

The symbol for square root √ is called the radical sign or radix.

The radicand is the number or expression under the radical sign √. For Example: 9 is radicand in \[\sqrt{9}\].
Solved Examples
Example 1: Find square root of 225 by repeated subtraction method.
Solution:
The steps are as follows:
Step 1: 225 – 1 = 224
Step 2: 224 – 3 = 221
Step 3: 221 – 5 = 216
Step 4: 216 – 7 = 209
Step 5: 209 – 9 = 200
Step 6: 200 – 11 = 189
Step 7: 189 – 13 = 176
Step 8: 176 – 15 = 161
Step 9: 161 – 17 = 144
Step 10: 144 – 19 = 125
Step 11: 125 – 21 = 104
Step 12: 104 – 23 = 81
Step 13: 81 – 25 = 56
Step 14: 56 – 27= 29
Step 15: 29 – 29 = 0
Here we got the result 0 in the 15th step, so the square root of 225 is 15 i.e \[\sqrt{225}\] = 15
Example 2:
Determine the square root of 169 by repeated subtraction method.
Solution:
The steps are as follows:
Step 1: 169 – 1 = 168
Step 2: 168 – 3 = 165
Step 3: 165 – 5 = 160
Step 4: 160 – 7 = 153
Step 5: 153 – 9 = 144
Step 6: 144 – 11 = 133
Step 7: 133 – 13 = 120
Step 8: 120 – 15 = 105
Step 9: 105 – 17 = 88
Step 10: 88 – 19 = 69
Step 11: 69 – 21 = 48
Step 12: 48 – 23 = 25
Step 13: 25 – 25 = 0
Here we got the result 0 in the 13th step, so the square root of 169 is 13 i.e \[\sqrt{169}\] = 13
Quiz Time

Find the square root of the given numbers by repeated subtraction method.

64 2. 121
1. What are the Properties of a Square Root Number?
Answer:
Properties of Square Root:

If a number has 2,3, 7 and 8 in the units place then it does not have a square root in natural numbers.

If a number ends in an odd number of zeros, then it does not have a square root in natural numbers.

The square root of an even number is even and that of an odd number is odd.

Negative numbers have no squares root in a set of real numbers.
2. What is Perfect Square?
Answer: A product a number with itself is said to be square or perfect square. Or in other words you can say that if any number is multiplied by itself it gives a perfect square.
For example 25 is a perfect square since it can be written as 81 = 9 x 9.
Perfect square is a number obtained by squaring two equal integers.
For testing if a given number is a perfect square or not we write the given number as the product of prime factors then we make pairs of the same factors.If there are factors all of which have a pair, then the given number is a perfect square.