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Observe the following:

2
2
2
Statement 1 = 1 2 = 4 3 = 9 4 = 16 5 = 25 6 = 36 7 = 49 8 = 64 9 = 81 10 = 100
2
2
2
2
2
2
2
Inference 1 = 1 4 = 2 9 = 3 16 = 4 25 = 5 36 = 6 49 = 7 64 = 8 81 = 9 100 = 10
Square root of natural numbers can be found by the following methods.
• Prime factorisation method
• Division method
Finding Square Root through Prime Factorisation
When a given number is a perfect square, we find the square root through prime factorisation
using the following steps.
Step 1: Find all the prime factors of the given number.
Step 2: Group the similar factors into pairs. 2 144

Step 3: Select one prime factor from each group and multiply them. Their product is 2 72
the square root of the given number. 2 36
2 18
Consider the number 144.
3 9
Let us find its square root using prime factorisation method. 3 3
We have 144 = 2 × 2 × 2 × 2 × 3 × 3 1

So, 144 = 2 × 2 × 3 = 12
Example 21: Find the square root by prime factorisation method. 2 7744
2 3872
(a) 7744 (b) 24336
2 1936
Solution: (a) Since, 7744 = 2 × 2 × 2 × 2 × 2 × 2 × 11 × 11. 2 968 2 24336

= (2 × 2) × (2 × 2) × (2 × 2) × (11 × 11) 2 484 2 12168
2 242 2 6084
\ 7744 = 2 × 2 × 2 × 11 = 88 11 121 2 3042
1521
3
(b) Since, 24336 = 2 × 2 × 2 × 2 × 3 × 3 × 13 × 13 11 11 3 507
1 13 169
= (2 × 2) × (2 × 2) × (3 × 3) × (13 × 13)
13 13
\ 24336 = 2 × 2 × 3 × 13 = 156 1

Example 22: Find the least number by which 2178 should be multiplied to make it a perfect square.
Also, find the square root of the perfect square so obtained. 2 2178
Solution: Resolve 2178 into prime factors, 3 1089
3 363
2
Now, 2178 = 2 × 3 × 11 2 11 121
11 11
Since, a perfect square has pairs of equal factors, here 2 is left unpaired. 1
Hence, the given number should be multiplied by 2 to get the resulting number a perfect square.
⇒ 2178 × 2 = 4356

2
2
2
2
And, 4356 = 2 × 3 × 11 = (2 × 3 × 11) = 66 2
Thus, 4356 = 66
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