Verification of Perfect Squares: Examples

Whether on the CBSE board or Gujarat board, perfect squares are a crucial fundamental concept in Mathematics that you must learn to build a strong foundation in the subject. Any number written as the product of an integer by itself or as the second exponent of an integer is referred to as a perfect square. The concept could seem confusing at first. However, a thorough understanding of this concept helps understand more complex calculations. To help you, we’ve explained what perfect squares are with the help of some examples in this article. We’ve also included properties for verification of perfect squares too.

What is a Perfect Square?

An integer multiplied by itself creates a perfect square, which is a positive integer. Perfect squares can be summed up as numbers that are the products of integers multiplied by themselves. Generally speaking, a perfect square can be expressed as x², where x is the integer and x² value, a perfect square. In other words, perfect squares are numbers that can be formed by squaring an integer or a whole number. Consider the examples given below;

Numbers 132 and 225 can be written as a product of their prime factors as;

132 = 2×2×3×11

225 = 3×3×5×5

Here,

There are no identical factors for this number. Thus, 132 cannot be a perfect square. The number 225, however, has pairs of identical factors.

225= 3×3×5×5 = 3² × 5² = (3 × 5) ²

Verification of Perfect Square; Properties

There are various properties that help you understand whether a number is a perfect square or not. Few of them are listed below.

Numbers ending in 2,3,7,8 will never be a perfect square.

Examples: 3562, 8787, 3253. All these numbers are not perfect squares.

Numbers ending with odd counts of zero are never perfect squares.

Examples: 3530, 38485000, 56700000

An even number’s square is always an even number.

Examples: = 3² = 16, 12²= 144, 6²= 36.

An odd number’s square is always an odd number.

Examples: 3²= 9, 11²= 121, 7²= 49

Natural numbers ?,? and c are considered a Pythagorean triplet if a²+ b² = c². For any number ?>1,(2?, n²–1, n²+1) is a Pythagorean triplet.

Example: for ?=4, (8,15,17) is a Pythagorean triplet.

Calculating the square root of the given integer is another method for determining whether a number is a perfect square or not. It is a perfect square if the square root is a whole number. A given number is not a perfect square if the square root is not a whole number.

For instance, let’s find the square root of 24 to see if it’s a perfect square or not. √24 Equals 4.89. As we can see, 24 is not a perfect square because 4.89 is not a whole number. Let’s use 81 as an example once more: 81 = 9. 81 is a perfect square because, as we can see, 9 is a whole number.

The unit digit in the square numbers that end in 3 and 7 will be 9, respectively.

Example: 7²= 49, 17²= 289

The unit digit in the square number of a number that ends in 5 will be 5.

Example: 15²= 225, 25²= 625

The units digit in the square number of a number that ends in 4 and 6 would be 6.

Example: 16²= 256, 24²= 576.

The unit digit in the square number of an integer that ends in 2 or 8 is 4.

Example: 8²= 64, 14²= 144

List of Perfect Squares

Given below is a table that lists the first 10 perfect squares that will help you solve the questions quickly.

Number	Square
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144

We’ve provided a brief overview of verification of perfect squares in this article. We hope the article was helpful. If you have any questions or queries, reach out to us.