Perfect Squares

Perfect squares are very interesting numbers that are important in mathematics and appear in geometry, algebra, and number theory making it an important concept in the study of mathematics. From simple squares of small numbers to complex ones, perfect squares help in understanding hidden interrelations of numbers and their roots. This article will help you explore the world of perfect squares to define them and detail their properties and their applications.

1.0Definition of a Perfect Square

A perfect square is a specific number expressed as the square of an integer. In other words, a number is a perfect square if we can write it in the form n2 , where n is a whole number. The example of a perfect square is, 

• 25 is an example of a perfect square because it can be expressed as $5^2$. 
• Similarly, 81 is a perfect square because $9^2$=81

2.0Perfect Squares 1 to 100 

Perfect squares from 1 to 100 are numbers that result from squaring whole numbers between 1 and 10. The complete list of perfect squares from 1 to 100 is:

3.0How to Find the Perfect Square

To determine whether a number is a perfect square, two methods can be used which includes: 

Square Root Method

1. Find the Square Root – Compute the square root of the number.
2. Check for a Whole Number Result – If the square root is a whole number, then the given number is a perfect square.
3. Verify with Squaring – Square the obtained integer and check if it equals the original number.

Let us try an example to understand this better. 

Example: Is 144 a perfect square?

• Find the square root: $\sqrt{144}=12$
• Check: 12 is a whole number, so 144 is a perfect square.
• Verify: $12^2=144$  confirming that 144 is a perfect square.

Prime Factorisation Method

• Step 1: Determine the prime factorisation of the number.
• Step 2: Verify if the exponents for all primes are even. Then, the number will be a perfect square. 

Example: Check whether 72 is a perfect square or not. 

• Factorise 72: we will get 72 = $2\times 2\times 2\times 3\times 3$
• Check: For the number to be a perfect square, all the exponents of the prime factors must be even; in the given case, one of the exponents of 2 is odd (3), hence, 72 is not a perfect square.  

4.0Perfect Squares Chart 

Below is a chart representing perfect squares from 1 to 40:

5.0Properties of Perfect Squares

Perfect squares possess some very distinct and interesting features that set them apart from most other numbers. Some of these key properties include:

• Square Root is an Integer: A perfect square always has an integer as its square root. For instance, the square root of 81 or 25 is 9 or 5, respectively, which is an integer. 
• Last Digit of a Perfect Square: The last digit of a perfect square will always be either 0, 1, 4, 5, 6, or 9. It cannot end in 2, 3, 7, or 8. For instance, 16 ends in 6 and 25 ends in 5, which are perfect squares.
• The sum of Consecutive Odd Numbers: A perfect square may be expressed as the sum of consecutive odd numbers starting from 1. For example:
• • 1 = 1
  • 1 +3 = 4
  • 1 + 3 + 5 = 9 
  • 1 + 3 + 5 + 7 = 16
• Difference between two perfect squares: The difference between two perfect squared numbers can be written in factored form as: (a+b)² = (a+b)(a–b)
• Perfect Squares are Non-negative: The squares are always non-negative. One can never find a perfect square that is negative since any positive or negative number, when squared, always produces a non-negative number. 

6.0Solved Examples

Problem 1: Find whether 225 is a perfect square.

Solution:

$\sqrt{225}=15,$ 

The square root of 225, which is a whole number 

Hence, 225 is a perfect square.

Problem 2: Find the perfect square of 23.

Solution: To find the perfect square of 23, simply multiply 23 by its own. We will get: $23^2=529$

Problem 3: Check whether 1000 is a perfect square.

Solution: By taking the square root of 1000, we will get, 

$\sqrt{1000}=31.62$ , which is not a whole number. 

Hence, 1000 is not a perfect square.