SQUARES OF NUMBERS

When a number is multiplied by itself we say that the number can be squared.

It is denoted by a number raised to the power 2.

For example :

(i) 3 ⋅ 3  =  3²

(ii) 5 ⋅ 5  =  5²

In example (ii) 5² is read as 5 to the power of 2 (or) 5 raised to the power 2 (or) 5 squared. 25 is known as the square of 5.

Similarly, 49 and 81 are the squares of 7 and 9 respectively.

In this section, we are going to learn a few methods of squaring numbers.

Perfect Square :

The numbers 1, 4, 9, 16, 25, g are called perfect squares or square numbers as

1 = 1², 4 = 2², 9 = 3², 16 = 4² and so on.

A number is called a perfect square if it is expressed as the square of a number.

Let us look into some example problems based on the above concept. 

Example 1 :

Find the perfect square numbers between

(i) 10 and 20 (ii) 50 and 60 (iii) 80 and 90.

Solution :

(i) The perfect square number between 10 and 20 is 16.

(ii) There is no perfect square number between 50 and 60.

(iii) The perfect square number between 80 and 90 is 81.

Example 2 :

By observing the unit’s digits, which of the numbers 3136, 867 and 4413 can not be perfect squares?

Solution :

Since 6 is in units place of 3136, there is a chance that it is a perfect square.

867 and 4413 are surely not perfect squares as 7 and 3 are the unit digit of these numbers

Example 3 :

Write down the unit digits of the squares of the following numbers:

(i) 24              (ii) 78          (iii) 35

Solution :

(i) The square of 24 = 24 × 24. Here 4 is in the unit place.

So, we have 4 ⋅ 4 = 16.

Hence 6 is in the unit digit of square of 24.

(ii) The square of 78 = 78 × 78. Here 8 is in the unit place.

So, we have 8 ⋅ 8  =  64

Hence 4 is in the unit digit of square of 78.

(ii) The square of 35 = 35 × 35. Here 5 is in the unit place.

So, we have 5 ⋅ 5  =  25

Hence 5 is in the unit digit of square of 35.

Example 4 :

Find the square of 3/8

Solution :

The square of (3/8)  =  (3/8)²

  =  (3/8) ⋅ (3/8)

  =  (3 ⋅ 3) / (8 ⋅ 8)

=  9/64 

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