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Square Root by Repeated Subtraction: Step-by-Step Guide

Square Root by Repeated Subtraction: Step-by-Step Guide

Reviewed by:

Rama Sharma

How to Find the Square Root Using 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, 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

Square Root of a Number	Number	Square Root of a Number	Number
$\sqrt{4}$	2	$\sqrt{196}$	14
$\sqrt{9}$	3	$\sqrt{225}$	15
$\sqrt{16}$	4	$\sqrt{256}$	16
$\sqrt{25}$	5	$\sqrt{289}$	17
$\sqrt{36}$	6	$\sqrt{324}$	18
$\sqrt{49}$	7	$\sqrt{361}$	19
$\sqrt{64}$	8	$\sqrt{400}$	20
$\sqrt{81}$	9	$\sqrt{441}$	21
$\sqrt{100}$	10	$\sqrt{484}$	22
$\sqrt{121}$	11	$\sqrt{529}$	23
$\sqrt{144}$	12	$\sqrt{576}$	24
$\sqrt{169}$	13	$\sqrt{625}$	25

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: Step-by-Step Guide

1. What is the repeated subtraction method for finding a square root?

The repeated subtraction method is a process used to find the square root of a perfect square. It works on the principle that the sum of the first 'n' consecutive odd numbers is equal to 'n²'. By successively subtracting consecutive odd numbers (1, 3, 5, 7, ...) from the given number until the result is zero, the number of subtractions performed gives you the square root.

2. How do you find the square root of 64 using the repeated subtraction method?

To find the square root of 64, you subtract consecutive odd numbers starting from 1 until you reach zero. The number of steps taken is the square root.

Step 1: 64 - 1 = 63
Step 2: 63 - 3 = 60
Step 3: 60 - 5 = 55
Step 4: 55 - 7 = 48
Step 5: 48 - 9 = 39
Step 6: 39 - 11 = 28
Step 7: 28 - 13 = 15
Step 8: 15 - 15 = 0

Since we performed the subtraction 8 times to reach zero, the square root of 64 is 8.

3. Why do we only subtract consecutive odd numbers in the repeated subtraction method?

This method is based on a fundamental property of square numbers: every square number is the sum of consecutive odd numbers starting from 1. For example, 4 (which is 2²) is 1 + 3. Similarly, 9 (which is 3²) is 1 + 3 + 5. By subtracting these odd numbers, we are essentially reversing the process of building the square. The count of the odd numbers subtracted to reach zero reveals the original number 'n' that was squared.

4. What are the main limitations of using the repeated subtraction method?

The repeated subtraction method has two significant limitations:

It only works for perfect squares. If you apply it to a non-perfect square (like 50), the subtraction will not end exactly at zero.
It is highly inefficient for large numbers. Finding the square root of a large number like 400 would require 20 steps of subtraction, which is time-consuming and prone to calculation errors. For larger numbers, methods like prime factorisation or long division are far more practical.

5. What happens if you try to find the square root of a non-perfect square, like 30, using this method?

If you apply the repeated subtraction method to a non-perfect square like 30, you will not be able to reach zero. The process would look like this:

30 - 1 = 29
29 - 3 = 26
26 - 5 = 21
21 - 7 = 14
14 - 9 = 5

At this point, the next odd number to subtract is 11, which is greater than the remainder of 5. Since the process does not end in zero, it proves that 30 is not a perfect square.

6. How does the repeated subtraction method compare to the prime factorisation method for finding square roots?

Both methods find the square root of perfect squares, but they work differently. The repeated subtraction method uses arithmetic (subtraction), while the prime factorisation method uses number theory. For small numbers, repeated subtraction can be quick. However, for larger numbers, prime factorisation is generally more efficient. It involves breaking the number into its prime factors and then pairing them up. For example, for 144, you would find its prime factors (2x2x2x2x3x3), pair them as (2x2)x(2x2)x(3x3), and take one factor from each pair (2x2x3) to get the root, which is 12. This is often faster than subtracting 12 times.