Perfect Square
A perfect square is a number that can be expressed as the product of two equal integers. For example, 25 is a perfect square because it is the product of two equal integers, 5 × 5 = 25. However, 21 is not a perfect square because it cannot be expressed as the product of two equal integers. (7 × 3 = 21). Let us learn more about perfect squares in this lesson.
1.  What are Perfect Squares? 
2.  First 20 Perfect Squares 
3.  How to Identify Perfect Squares? 
4.  FAQs on Perfect Squares 
What are Perfect Squares?
Perfect squares are numbers which are obtained by squaring a whole number. Let us look at an example to understand the concept behind perfect squares. For this, we can take a set of 4 marbles and another set of 6 marbles. Let us arrange the marbles. Were you able to arrange the marbles in the way as shown below?
Let's analyze this activity. We can form a square with 4 marbles such that there are 2 rows, with 2 marbles in each row. With 6 marbles, we can form a rectangle such that there are 2 rows, with 3 marbles in each row. Mathematically, it means 4 = 2 × 2 and 6 = 3 × 2. Let us focus only on the numbers which form a square. Here, 4 = 2 × 2 = 2^{2}. Now, if we look at the definition of a perfect square, it says, "A perfect square is a number which is obtained by squaring a whole number. Let us assume if N is a perfect square of a whole number x, this can be written as N = the product of x and x = x^{2}. So, the perfect square formula can be expressed as:
Let us substitute the formula with values. If x = 9, and N = x^{2}. This means, N = 9^{2} = 81. Here, 81 is a perfect square because it is the square of a whole number. This can be understood in another way with the help of square roots. To know whether a number is a perfect square or not, we calculate the square root of the given number. If the square root is a whole number, then the given number is a perfect square, but if the square root value is not a whole number, then the given number is not a perfect square. For example, to check whether 21 is a perfect square or not, let us calculate its square root. √21 = 4.58. As we can see, 4.58 is not a whole number, so, 21 is not a perfect square. Let us take another example of the number 64. √64 = 8. We can see that 8 is a whole number, therefore, 64 is a perfect square. Now, that we know about perfect squares, let's learn about perfect square trinomials.
Perfect Square Trinomial: An expression which is obtained from the square of a binomial equation is termed as a perfect square trinomial. For example, if we square the expression (y+3), we use the identity, (a+b)^{2}= a^{2} +2ab+b^{2}, and we get, (y+3)^{2} = y^{2} +6y+9. Here, y^{2} +6y+9 is a perfect square trinomial. Other examples of a perfect square trinomial are y^{2} 8y+16 and 4x^{2}+ 12x +9.
First 20 Perfect Squares
The table given below shows the perfect squares of the first 20 natural numbers. The first column shows the natural number and the second column shows the square of the natural number. You can easily find the square of a natural number by multiplying it by the number itself. For example, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, and so on.
Natural 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 
13  169 
14  196 
15  225 
16  256 
17  289 
18  324 
19  361 
20  400 
How to Identify Perfect Squares?
Observe the last digit of the perfect squares of numbers 1 to 20 as given in the table above. You will notice that they end with any one of these digits 0, 1, 4, 5, 6, or 9. After trying various perfect square numbers you would have observed an important property of perfect squares. Numbers that have any of the digits 2, 3, 7, or 8 in their units place are nonperfect squares, whereas, numbers that have any of the digits 0, 1, 4, 5, 6, or 9 in their units place are perfect squares. The following observations can be made to identify a perfect square.
 The numbers ending with 3 and 7 will have 9 as the unit digit in its square number.
 The number ending with 5 will have 5 as the unit digit in its square number.
 The number ending with 4 and 6 will have 6 as the unit digit in its square number.
 The number ending with 2 and 8 will have 4 as the unit digit in its square number.
 The numbers ending with 1 and 9 will have 1 as the unit digit in its square number.
Let us look at a few deviations from these abovedefined rules of a perfect square number.The numbers 159 and 169 both end with the digit 9 but 169 is a perfect square, whereas 159 is not. If the number ends with the digit 0, then you may look for the following: How many zeros are there at the end of the number? Let's say we have a number 1000. If there are odd number of zeros, then it's definitely not a perfect square. 1000 has 3 zeros at the end. Thus, it's not a perfect square. If there are even number of zeros, then it is a perfect square. 400 and 300 both have even number of zeros at the end, but 400 = 20^{2}, which is a perfect square, but 300 is not a square of any whole number.
Another Way to Identify Perfect Squares
Another way to check whether a number is a perfect square or not, we calculate the square root of the given number. If the square root is a whole number, then it is a perfect square. If the square root is not a whole number, then the given number is not a perfect square. For example, to check whether 24 is a perfect square or not, let us calculate its square root. √24 = 4.89. As we can see, 4.89 is not a whole number, so, 24 is not a perfect square. Let us take another example of the number 81. √81 = 9. We can see that 9 is a whole number, therefore, 81 is a perfect square.
A Quick Way to Square Special Numbers
You can find the square of a number by multiplying it with itself, for example, 6 × 6 = 36, However, there are some simple methods which work for special types of numbers. These can be applied to square a number in a very short time. In other words, this can be used to calculate the square of a large number without the long process of multiplication.
Numbers Ending with Digit 5: Let's consider a number ending with 5, like, 65. Now, we can find the square of 65 through a sequence of four simple steps. First, we need to separate the numbers 6 and 5. Next, multiply 6 with its successor 7. Now for the third step, square the number 5 to get 25. Further, for the final step write the digits of the second step, followed by 25. The final answer for the square of 65 is 4225.
Decimal Numbers: In order to find the square of a decimal number, we need to remember a simple rule. Rule: If there are n number of digits after the decimal in a decimal number, then the number of digits after the decimal in its perfect square will be 2n. Let us look at an example to understand this better. We know that 8^{2}=64, thus, 0.08^{2} =0.0064. There are 2 digits after the decimal in 0.08. Hence, there will be 2 × 2 = 4 digits after the decimal in its square, which is 0.0064.
Important Notes
The following important points should be kept in mind while we work with perfect squares.
 A perfect square that ends with 0 will always have an even number of zeros at the end.
 Perfect squares are always positive as (ve) × (ve) = (+ve).
 The square roots of perfect squares may be positive or negative.
 We can also find perfect cubes by multiplying a number with itself thrice.
 To check whether a given number is a perfect square or not, we can calculate the square root of the given number. If the square root is a whole number, then it is a perfect square. If the square root is not a whole number, then the given number is not a perfect square.
Solved Examples

Example 1: In an auditorium, the number of rows are the same as the number of columns. If there are 60 chairs in a row, how many chairs are there in the auditorium?
Solution:
It is given that the number of rows are the same as the number of columns. This indicates that the chairs are arranged in the form of a square. To find the total number of chairs in the auditorium, we will find the square of 60 units as a side. 60^{2}= 60 × 60 =3600. Therefore, there are 3600 chairs in the auditorium.

Example 2: What will be the area of the square of side of length ( y + 2) inches?
Solution:
The given figure is a square with each side equal to (y+2) inches. Thus, using perfect squares formula for trinomials we get Area = ( y+2)^{2} = y^{2}+(2 × y × 2) +2^{2 }= y^{2} + 4y + 4. Therefore, the area of the given figure is (y^{2} + 4y + 4) square inches.
FAQs on Perfect Squares
How Can you Tell if a Number is a Perfect Square?
A number is considered to be a perfect square if it can be written as a square of a whole number. For example, 9 is a perfect square because 3 × 3 = 3^{2 }= 9. However, 21 is not a perfect square, because there is no whole number which can be squared to give 21 as the product. (7 × 3 = 21)
What is Perfect Squares Trinomial?
An expression which is obtained from the square of a binomial equation is termed as a perfect square trinomial. For example, (x3)^{2} =(x^{2} 6x+9). Here, (x^{2} 6x+9) is a perfect trinomial.
What is a Perfect Cube?
If a number X can be expressed in the form X = a × a × a, then X is termed as a perfect cube. For example, 8 = 2^{3}, here 8 is a perfect cube because 2 × 2 × 2 = 8.
What Numbers are Perfect Squares?
The numbers which can be written as a product of two equal numbers can be referred to as perfect squares. A few examples of perfect square numbers are 49, 64, 81,100.
How to Factor Perfect Squares?
A perfect square number can be factorized just as we factorize a normal number. It can be written as a product of two equal numbers. For example, the number 16 can be factorized as 4 × 4, or it can be factorized as a product of prime numbers. 16 = 2 × 2 × 2 × 2.
What are the Perfect Squares Between 1 and 100?
There are ten perfect squares between 1 and 100. They can be listed as 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.