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SQUARES AND SQUARE ROOTS 97

6.4 Finding the Square of a Number
Squares of small numbers like 3, 4, 5, 6, 7, ... etc. are easy to find. But can we find the
square of 23 so quickly?
The answer is not so easy and we may need to multiply 23 by 23.

There is a way to find this without having to multiply 23 × 23.
We know 23 = 20 + 3
2
2
Therefore 23 = (20 + 3) = 20(20 + 3) + 3(20 + 3)
2
= 20 + 20 × 3 + 3 × 20 + 3 2
= 400 + 60 + 60 + 9 = 529
Example 1: Find the square of the following numbers without actual multiplication.
(i) 39 (ii) 42

Solution: (i) 39 = (30 + 9) = 30(30 + 9) + 9(30 + 9)
2
2
2
= 30 + 30 × 9 + 9 × 30 + 9 2
= 900 + 270 + 270 + 81 = 1521
(ii) 42 = (40 + 2) = 40(40 + 2) + 2(40 + 2)
2
2
2
= 40 + 40 × 2 + 2 × 40 + 2 2
= 1600 + 80 + 80 + 4 = 1764
6.4.1 Other patterns in squares
Consider the following pattern:
2
25 = 625 = (2 × 3) hundreds + 25
Consider a number with unit digit 5, i.e., a5
2
35 = 1225 = (3 × 4) hundreds + 25
(a5) 2 = (10a + 5) 2
2
75 = 5625 = (7 × 8) hundreds + 25
= 10a(10a + 5) + 5(10a + 5)
2
125 = 15625 = (12 × 13) hundreds + 25
= 100a + 50a + 50a + 25
2
Now can you find the square of 95? = 100a(a + 1) + 25
= a(a + 1) hundred + 25
TRY THESE
Find the squares of the following numbers containing 5 in unit’s place.
(i) 15 (ii) 95 (iii) 105 (iv) 205

6.4.2 Pythagorean triplets
Consider the following
2
2
3 + 4 = 9 + 16 = 25 = 5 2
The collection of numbers 3, 4 and 5 is known as Pythagorean triplet. 6, 8, 10 is
also a Pythagorean triplet, since
2
2
6 + 8 = 36 + 64 = 100 = 10 2
Again, observe that
2
2
5 + 12 = 25 + 144 = 169 = 13 . The numbers 5, 12, 13 form another such triplet.
2