Perfect Square

A perfect square is a number that can be expressed as the product of an integer by itself or as the second exponent of an integer. For example, 25 is a perfect square because it is the product of integer 5 by itself, 5 × 5 = 25. However, 21 is not a perfect square number because it cannot be expressed as the product of two same integers.

In this article, we will discuss the concept of perfect squares and learn how to identify them. We will discuss the definition of a perfect square, its formula, and the list of perfect squares along with a few solved examples for a better understanding. 

A perfect square is a positive integer that is obtained by multiplying an integer by itself. In simple words, we can say that perfect squares are numbers that are the products of integers by themselves. Generally, we can express a perfect square as x², where x is an integer and the value of x² is a perfect square. 

Perfect squares are numbers that are obtained by squaring a whole number or an integer. 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 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 = 2². Now, if we look at the definition of a perfect square, it says, "A perfect square is a number which is obtained by squaring an integer."

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 = x². So, the perfect square formula can be expressed as:

Let us substitute the formula with values. If x = 9, and N = x². This means, N = 9² = 81. Here, 81 is a perfect square because it is the square of a whole number, 9. 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 number. 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/integer, so, 21 is not a perfect square number. 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 that is obtained from the square of a binomial is termed a perfect square trinomial. For example, if we square the expression (y+3), we use the identity, (a+b)²= a² +2ab+b², and we get, (y+3)² = y² +6y+9. Here, y² +6y+9 is a perfect square trinomial. Other examples of a perfect square trinomial are y² -8y+16 and 4x²+ 12x +9.

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 itself. For example, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, and so on.

Look at these lists of perfect squares 1-100 to have a better understanding of perfect square numbers 1 to 100.

• Perfect squares 1 to 20
• Perfect squares 1 to 30
• Perfect squares 1 to 40
• Perfect squares 1 to 50
• Perfect squares 1 to 100

Observe the last digit of the perfect square 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 non-perfect square numbers, whereas, numbers that have any of the digits 0, 1, 4, 5, 6, or 9 in their units place might be 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 units place digit in its square number.
• The number ending with 5 will have 5 as its units place digit in its square number.
• The number ending with 4 and 6 will have 6 as the units place digit in its square number.
• The number ending with 2 and 8 will have 4 as the units place digit in its square number.
• The numbers ending with 1 and 9 will have 1 as the units place digit in its square number.

Let us look at a few deviations from the above-defined 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 is an 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 an even number of zeros, then it might be a perfect square. 400 and 300 both have an even number of zeros at the end, but 400 = 20², 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 is by calculating 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.

You can find the square of a number by multiplying it by itself, for example, 6 × 6 = 36, However, there are some simple methods that 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 using the long multiplication method.

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 by its successor, i.e. 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.

Perfect Square Tips and Tricks:

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 by 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.

► Related Articles

• Perfect Square Formula
• Perfect Square Trinomial Formula
• Square Number

What is a Perfect Square?

A perfect square is a number that is the second exponent of an integer. For example, let us take any integer, 'a'. The perfect square will be a × a, or a².

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 an integer. For example, 9 is a perfect square because 3 × 3 = 3² = 9. However, 21 is not a perfect square, because there is no whole number that can be squared to give 21 as the product.

What is a Perfect Square Trinomial?

An expression that is obtained from the square of a binomial is termed a perfect square trinomial. For example, (x-3)² =(x² -6x+9). Here, (x² -6x+9) is a perfect square trinomial.

What Numbers are Perfect Squares?

The numbers which can be written as a product of a number by itself can be referred to as perfect squares. A few examples of perfect square numbers are 49, 64, 81, and 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 a number by itself. For example, the number 16 can be factorized as 4 × 4, or it can be factorized as a product of prime numbers as 16 = 2 × 2 × 2 × 2.

What are the Perfect Squares Between 1 and 100?

There are ten perfect squares from 1 and 100. They can be listed as 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

How to Find Perfect Square?

To find a perfect square, we need to multiply the whole number by itself. The first 20 perfect square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, and 400.

Is 7 a Perfect Square?

A perfect square is a number that can be expressed as a product of a whole number by itself. The factors of 7 are 1 and 7 only. So, we cannot express 7 as a product of any integer/whole number. So, 7 is not a whole number.