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Example 1: Find the square of the following numbers:

(a) 16 (b) –10 (c) 14
Solution: (a) Square of 16 = 16 = 16 × 16 = 256
2
2
(b) Square of –10 = (–10) = (–10) × (–10) = 100
(c) Square of 14 = 14 = 14 × 14 = 196
2
Square of a Rational Number

The square of a rational number is obtained by multiplying the rational number two times or
squaring its numerator and denominator separately.

×
×
4
4
4
For example, (a)    2 =  4 ×    = 44 = 16 or    2 = 4 2 2 = 44 = 16
 
5

5

5


5
25
×
25
55
5
55
×
5
−  5
5
25
−  5
−  5
25
(b)    11   2 =    11  ×  −  5   = −× −5 = 121 or    11   2 = − ( ) 5 2 2 = − ( ) ×− ( ) 5 = 121


× 11
× 11
11
11
 11
11
2
Note: • The square of an integer is always a whole number. For example, (–5) = 25 (Whole number)
• The square of a proper fraction is smaller than itself.
 3 2 3 3 9 3
For example,   = 4 × 4 = 16 < 4

4
Perfect Squares
The squares of first twenty natural numbers has been given in the table. Observe it carefully.
Number 1 2 3 4 5 6 7 8 9 10
Square 1 4 9 16 25 36 49 64 81 100
Number 11 12 13 14 15 16 17 18 19 20
Square 121 144 169 196 225 256 289 324 361 400
Numbers like 9 = 3 × 3, 16 = 4 × 4, 225 = 15 × 15, 400 = 20 × 20 are called square numbers or perfect
squares because they are squares of natural numbers.
A natural number n is considered a perfect square if there exists some natural number m such
2
that n = m × m = m .
Natural numbers such as 12, 15, 24 and 35 are not perfect squares.
There are non-perfect square numbers in between two consecutive square numbers. For example,
5, 6, 7 and 8 which lie between perfect squares 4 and 9, are non-perfect square numbers.
How do we know whether a given natural number is a perfect square?
For this, we follow these steps:
Step 1: Write the number as a product of its prime factors.
Step 2: Make pairs of the same prime factors.

Step 3: Check for the unpaired factor.

123 Squares and Square Roots