²

**Properties of perfect square :**

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

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

For example,

(i) 3 x 3 = 3² = 9

(ii) 5 x 5 = 5² = 25

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.

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.**

We observe the following properties through the patterns of perfect squares.

**Property 1 : **

In perfect squares, the digits at the unit’s place are always 0, 1, 4, 5, 6 or 9. The numbers having 2, 3, 7 or 8 at its units' place are not perfect square numbers.

**Property 2 :**

If a number has 1 or 9 in the unit's place then its square ends in 1.

**Property 3 :**

If a number has 2 or 8 in the unit's place then its square ends in 4.

**Property 4 :**

If a number has 3 or 7 in the unit's place then its square ends in 9.

**Property 5 :**

If a number has 4 or 6 in the unit's place then its square ends in 6.

**Property 6 :**

If a number has 5 in the unit's place then its square ends in 5.

**Property 7 :**

Consider the following square numbers :

From the perfect squares given above, we infer that

**Property 8 :**

Consider the following square numbers :

Therefore, 100 is a perfect square and 81000 is not a perfect square.

**Property 9 :**

Square of even numbers is always even.

It has been illustrated in the table given below.

**Property 10 :**

Square of odd numbers is always odd.

It has been illustrated in the table given below.

From property 11 and property 12, we infer that

Addition of consecutive odd numbers :

The above figure illustrates the result that the sum of the first "n" natural odd numbers is n².

And, square of a rational number "a/b" is given by

After having gone through the stuff given above, we hope that the students would have understood "Perfect square".

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