How to Find Square Root Using Repeated Subtraction with Steps and Solved Examples
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, the square of 4 is 16 and the square root of 16 is 4. Squaring 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 the 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 \[\sqrt{121}\]= 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
FAQs on Square Root by Repeated Subtraction Method Explained
1. What is square root by repeated subtraction?
The square root by repeated subtraction method is a technique to find the square root of a perfect square by subtracting consecutive odd numbers from it until the result becomes zero. The number of subtractions performed gives the square root.
- Subtract 1, then 3, then 5, then 7, and so on.
- Continue subtracting consecutive odd numbers.
- When the result becomes 0, count the total subtractions.
- The count equals the square root of the number.
2. How do you find the square root by repeated subtraction?
To find the square root by repeated subtraction, subtract consecutive odd numbers from the given number until you reach zero. Follow these steps:
- Step 1: Start with the given perfect square (e.g., 25).
- Step 2: Subtract 1 → 25 − 1 = 24.
- Step 3: Subtract 3 → 24 − 3 = 21.
- Step 4: Subtract 5 → 21 − 5 = 16.
- Step 5: Subtract 7 → 16 − 7 = 9.
- Step 6: Subtract 9 → 9 − 9 = 0.
3. Why does the repeated subtraction method work for square roots?
The repeated subtraction method works because the sum of the first n odd numbers equals n². In other words:
- 1 = 1²
- 1 + 3 = 4 = 2²
- 1 + 3 + 5 = 9 = 3²
- 1 + 3 + 5 + 7 = 16 = 4²
4. Can you give an example of finding a square root by repeated subtraction?
Yes, for example, the square root of 16 can be found by repeated subtraction of odd numbers until zero is reached.
- 16 − 1 = 15
- 15 − 3 = 12
- 12 − 5 = 7
- 7 − 7 = 0
5. Does the repeated subtraction method work for non-perfect squares?
No, the repeated subtraction method works exactly only for perfect squares. If the number is not a perfect square, the subtraction process will not end exactly at zero.
- Example: For 20, subtracting 1, 3, 5, 7, 9 gives a negative number before reaching zero.
- This shows that 20 is not a perfect square.
6. What is the formula behind square root by repeated subtraction?
The formula behind square root by repeated subtraction is based on 1 + 3 + 5 + … + (2n − 1) = n². This means:
- The sum of the first n odd numbers equals n squared.
- If a number can be reduced to zero by subtracting n consecutive odd numbers, then its square root is n.
7. How many times do you subtract to get the square root?
The number of times you subtract consecutive odd numbers until reaching zero equals the square root of the number. For example:
- For 36:
- 36 − 1 − 3 − 5 − 7 − 9 − 11 = 0
- Total subtractions = 6
8. What type of numbers can be solved using the repeated subtraction method?
The repeated subtraction method can be used for perfect square natural numbers. These include numbers like:
- 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.
9. What are the advantages of finding square roots by repeated subtraction?
The main advantage of the repeated subtraction method is that it clearly shows the relationship between odd numbers and perfect squares. Key benefits include:
- Easy to understand for beginners.
- Helps verify perfect squares.
- Strengthens conceptual understanding of square numbers.
10. What is the difference between square root by repeated subtraction and prime factorization?
The main difference is that repeated subtraction uses consecutive odd numbers, while prime factorization uses prime factors to find the square root.
- Repeated subtraction counts how many odd numbers reduce the number to zero.
- Prime factorization breaks the number into prime factors and pairs identical factors.
- Prime factorization is faster for large perfect squares.
