Perfect Squares

You may hardly come across a mathematical problem that doesn’t involve the use of squares or perfect cubes. The use of exponents and powers in mathematics is introduced at the initial levels like class 6th or 7th and has applications at higher-level mathematics as well. Be it any of the competitive exams like GMAT, CAT, SSC CGL, etc., it is important to have a stronghold over this topic as it enhances your mental maths, thereby aiding in faster calculations. Through this blog, we will list down perfect squares and will also highlight their properties and important tricks to remember them.  

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What are Perfect Squares? 

The numbers which express the product of two equal integers are referred to as Perfect Squares. In simpler words, a perfect square is a number that you get when you multiply two same numbers. For example, The perfect square of 2 will be: 2^2= 4 

So, 4 is the perfect square of 2 as we get it, 2 x 2= 4 

List of Perfect Squares 

Be it university-level mathematics or Maths for competitive exams, it is essential to learn squares at least up to 25 as some questions may require complex calculations. Once you are through with the squares up to 25, you can explore them further. Tabulated below is the list of squares in the integer form. 

Number	Squares	Number	Squares
1	1	-1	1
2	4	-2	4
3	9	-3	9
4	16	-4	16
5	25	-5	25
6	36	-6	36
7	49	-7	49
8	64	-8	64
9	81	-9	81
10	100	-10	100
11	121	-11	121
12	144	-12	144
13	169	-13	169
14	196	-14	196
15	225	-15	225
16	256	-16	256
17	289	-17	289
18	324	-18	324
19	361	-19	361
20	400	-20	400
21	441	-21	441
22	484	-22	484
23	529	-23	529
24	576	-24	576
25	625	-25	625

Quick Fact: You may have noticed in the aforementioned table that negative numbers also have the same squares. This is because of the fact that upon multiplying 2 negative’s, the final product becomes positive. For example: (-8) × (-8) = 64 

Properties of These Numbers 

Perfect squares tend to showcase some similar patterns, hence, they have similar properties. Let us have a look at them and understand their properties through examples.  

Property 1

All the numbers which end in 2, 3, 7, or 8 are never a perfect square of a number. 

Property 2

The square of an even number is always an even number. 

For Example: 2^2= 4, 4^4= 16, etc. 

Property 3 

A proper fraction will always have its square root smaller to its original value. 

For Example– (⅔)^2 = 4/9

Property 4 

If a number consists of an odd number of zeros at the end, then such numbers can’t be perfect squares. 

For Example: 40, 16000, 25000 etc

Solved Examples of Perfect Squares 

Q1. Check whether 441 is a perfect square.

Sol: We can check by starting with the factorization method, 

441 = 3 x 3 x 7 x 7 

Both numbers exist two times, we can multiply both of them 

21 x 21 = (21)2

Hence, 441 is a perfect square. 

Q2. The area of a square dining table is 3600 cm2. Calculate the length of one side of the table. 

Sol: Length can be determined by the square of the area: 

3600  = 60 x 60

The length of the table is 60cms. 

Q3. Calculate the length of a square flower garden. The area is 81 square yards.

Sample Questions 

Let us quickly solve the questions listed below keeping in mind the aforementioned properties and list of perfect squares. 

1. Calculate the square root of 0.036
2. Determine whether 67436 is a perfect square or not?
3. Explain how 200 is not a perfect square. 
4. What will be the smallest number by which 387 must be multiplied to make it a perfect square
5. Determine the smallest number which must be added to 583 to make it a perfect square.
6. If x is subtracted from 1700 it becomes a perfect square. What is the value of x?
7. There are 8 odd numbers. Describe how the sum of these 8 numbers is 64. 
8. Find out the answer of (141)² – (140)². 
9. Determine the final value of the number 151 
10.  Which amongst the following is the square root of the number 64009?
    
    a. 357
    b. 253
    c. 412
    d. 833

Worksheet on Perfect Squares

Hopefully, through this blog, you are now familiar with the concept of perfect squares. Apart from this, exams like GMAT, CAT, etc, consist of various aptitude-based questions. To score well, thorough practice is required. We, at Leverage Edu, offer engaging online classes through Leverage Live wherein the experts will familiarize you with a plethora of similar concepts and you will also get to practice a wide range of questions. Hurry up! Book an online demo session now! Limited seats left!