Squares and Cubes
Squares and Cubes are mathematical operations involving numbers that are essential in various areas of mathematics. Square of a number is obtained by multiplying the number by itself where as a cube of a number is obtained by multiplying the number by itself twice.
In this article, we will learn what is Square and Cube Number. We will also learn about Perfect Squares and Cube and Square and Cube charts 1 to 100.
Table of Content
- What is Square Number?
- What is Cube Number?
- Perfect Square and Cubes
- Squares and Cubes from 1 to 50
- Chart of Squares and Cubes 1 to 100
What is Square of a Number?
When an integer is multiplied by itself, it is called a square of that number. In simple words, a number that is multiplied two times is known as a square number. A square number is denoted as βn2 β in mathematics.
Examples of Square Numbers:
Suppose a number β7β is given. To find its square, just multiply it again by β7β. Here, we get 7β¨―7= 49. So, β49β is the square of β7β. Some more examples of finding a square number are below:
- 52 = 5β¨―5= 25
- 122 = 12β¨―12= 144
What is Cube of a Number?
When we multiply an integer three times by itself, it is called a cube of that number. In other words, when an integer is multiplied by its square, it becomes a cube number. It is denoted as βn3 β in mathematics.
Examples of Cube Numbers
Let us take an integer β3β. First find its square number 3β¨―3= 9. Now, multiply β9β with 3 again 9β¨―3= 27. Here, β27β is called the cube of β3β.
Also, we can simply multiply it thrice to find its cube. Suppose a number β6β. Multiply it three times by itself 6β¨―6β¨―6= 216. The cube of β6β is β216β. Some more example are as follows:
- 23 = 8
- 113 = 1331
Square and Cube 1 to 20
In this, we will learn the square and cube of numbers from 1 to 20. Letβs have a look on them
Square 1 to 20
Number | Square | Number | Square |
---|---|---|---|
1 | 1 | 11 | 121 |
2 | 4 | 12 | 144 |
3 | 9 | 13 | 169 |
4 | 16 | 14 | 196 |
5 | 25 | 15 | 225 |
6 | 36 | 16 | 256 |
7 | 49 | 17 | 289 |
8 | 64 | 18 | 324 |
9 | 81 | 19 | 361 |
10 | 100 | 20 | 400 |
Cube 1 to 20
Number | Cube | Number | Cube |
---|---|---|---|
1 | 1 | 11 | 1331 |
2 | 8 | 12 | 1728 |
3 | 27 | 13 | 2197 |
4 | 64 | 14 | 2744 |
5 | 125 | 15 | 3375 |
6 | 216 | 16 | 4096 |
7 | 343 | 17 | 4913 |
8 | 512 | 18 | 5832 |
9 | 729 | 19 | 6859 |
10 | 1000 | 20 | 8000 |
Squares and Cubes from 1 to 30
Squares and cubes of any number are very important for solving complex mathematical problems. They provide a basic idea to evaluate a question. Every student should memorize the squares and cubes from 1 to 30 as these serve as the basic pillars for simplifying problems.
Table of Squares and Cubes (1 to 30)
In this section, we will learn the square and cubes from 1 to 30. This will help students to solve the problems related to arithmetic operations. For any student, these are the basic squares and cubes which helps them to calculate easily and quickly. Here is the table in which the squares and cubes from 1 to 30 are given:
Perfect Square and Cubes
Perfect Squares and Cubes are those numbers whose square root and cube root is a natural number or an intger in case of cube. Not every number we come across is a perfect square or cube. Hence, we need to learn what are perfect squares and cubes and also learn how to check a perfect square or cube.
Perfect Square
A perfect square is the square of an integer that can be multiplied by itself two times. For example β16β is a perfect square because 4β¨―4= 16.
Perfect Cube
A perfect cube is the cube of an integer that can be multiplied by itself three times. For example, β27β is a perfect cube because 3β¨―3β¨―3= 27.
How to Identify Perfect Squares and Cubes?
After learning the definition of both perfect squares and perfect cube, we learn some easy ways for their identification. First, we learn about βunit digitβ or βend digitβ method, then we study another method prime factorization:
Unit Digit Method
Unit Digit Method is helpful in knowing about the possibility of a number being perfect square or cube without any actual test just by looking at the unit digit of a number. Letβs learn more about it.
- Squares: If a number is a perfect square, its end digit should only be 0, 1, 4, 5, 6, or 9. Any square whose unit digit is any other digit cannot be a perfect square. For example, the square of β7β is β49β. Here, 49 is a perfect square because its unit digit is β9β. Similarly, take the square of β12β. The square of β12β is β144β which is a perfect square, because its unit digit is β4β.
- Cubes: The unit digit of a perfect cube should only be 0, 1, 8, or 9. In this way, we can easily verify whether a number is a perfect cube or not. For example, β1728β is a perfect cube because its last digit is β8β.
Note: There are also some exceptions. Some numbers are both a βperfect squareβ and a βperfect cubeβ. For example, β64β is both perfect square and perfect cube.
Prime Factorization Method
Since, Unit digit method only gives a hint about the possibility of a number being perfect square or cube. However, the actual clarity can be gained only through prime factorization method.
- Squares: In this method, when we prime factorize any number, each group of prime factors should have an even exponent i.e. βmultiples of 2β². For example, the given number is β36β. Its prime factor is 22 β¨― 32 . Each group of prime factors has an even exponent β2β therefore 36 is a perfect square.
- Cubes: In this method, when we prime factorize any number, each group of prime factors should have an exponent that is multiple of β3β. For example, the given number is β64β. Its prime factor is 26 . The exponent β6β is multiple of β3β therefore 64 is a perfect cube.
Squares and Cubes from 1 to 50
Here, a list of squares and cubes from 1 to 50 is given. Learning these values will help students to reduce their calculation time and they can easily solve complex problems.
Squares 1 to 50
Number | Square | Number | Square | Number | Square | Number | Square | Number | Square |
---|---|---|---|---|---|---|---|---|---|
1 | 1 | 11 | 121 | 21 | 441 | 31 | 961 | 41 | 1681 |
2 | 4 | 12 | 144 | 22 | 484 | 32 | 1024 | 42 | 1764 |
3 | 9 | 13 | 169 | 23 | 529 | 33 | 1089 | 43 | 1849 |
4 | 16 | 14 | 196 | 24 | 576 | 34 | 1156 | 44 | 1936 |
5 | 25 | 15 | 225 | 25 | 625 | 35 | 1225 | 45 | 2025 |
6 | 36 | 16 | 256 | 26 | 676 | 36 | 1296 | 46 | 2116 |
7 | 49 | 17 | 289 | 27 | 729 | 37 | 1369 | 47 | 2209 |
8 | 64 | 18 | 324 | 28 | 784 | 38 | 1444 | 48 | 2304 |
9 | 81 | 19 | 361 | 29 | 841 | 39 | 1521 | 49 | 2401 |
10 | 100 | 20 | 400 | 30 | 900 | 40 | 1600 | 50 | 2500 |
Cube 1 to 50
Number | Cube | Number | Cube | Number | Cube | Number | Cube | Number | Cube |
---|---|---|---|---|---|---|---|---|---|
1 | 1 | 11 | 1331 | 21 | 9261 | 31 | 29791 | 41 | 68921 |
2 | 8 | 12 | 1728 | 22 | 10648 | 32 | 32768 | 42 | 74088 |
3 | 27 | 13 | 2197 | 23 | 12167 | 33 | 35937 | 43 | 79507 |
4 | 64 | 14 | 2744 | 24 | 13824 | 34 | 39304 | 44 | 85184 |
5 | 125 | 15 | 3375 | 25 | 15625 | 35 | 42875 | 45 | 91125 |
6 | 216 | 16 | 4096 | 26 | 17576 | 36 | 46656 | 46 | 97336 |
7 | 343 | 17 | 4913 | 27 | 19683 | 37 | 50653 | 47 | 103823 |
8 | 512 | 18 | 5832 | 28 | 21952 | 38 | 54872 | 48 | 110592 |
9 | 729 | 19 | 6859 | 29 | 24389 | 39 | 59319 | 49 | 117649 |
10 | 1000 | 20 | 8000 | 30 | 27000 | 40 | 64000 | 50 | 125000 |
Patterns in Squares and Cubes
There are some interesting patterns in squares and cubes which often show some distinct properties and mathematical relations. Here are some important patterns which every student should know:
Patterns in Square Numbers
There are various patterns in the square numbers, some of those patterns are:
- Difference between square numbers
The difference between any two consecutive squares is always an odd number. For example, Two consecutive squares β4β and β9β are given. Their difference is 9-4= 5, which is an odd number.
- Sum of consecutive natural numbers
Whenever we square any odd number, the resultant will always be the sum of two consecutive natural numbers. Suppose we take the square of β3β which is β9β. Here, β9β is the result of the addition of two consecutive numbers β4β and β5β.
- Product of two consecutive even or odd natural numbers
The product of two consecutive even numbers or consecutive odd numbers is also an important pattern of square numbers. For example, β25β is the product of odd numbers 5β¨―5. Similarly, β64β is the product of two even numbers 8β¨―8.
- Adding first n odd numbers
A square of any number is obtained by the sum of first βnβ odd numbers. Suppose, a number is given β25β. Here, 25 is obtained by the addition of the first 5 odd numbers i.e. (1+3+5+7+9).
- Adding triangular numbers
Triangular numbers are the numbers obtained by adding the next natural number and it forms an equilateral triangle. The formula to find triangular numbers is:
T(n)= 1+2+3+4β¦β¦.+n
Now, by adding these triangular numbers, square numbers can be generated easily. For example, the square number is β4β, which is the addition of the first triangular number to itself i.e. (1+3).
Patterns in Cube Numbers
Some commons patterns in cubes are:
- Adding consecutive odd numbers
By adding consecutive odd numbers, we can easily find the next cube numbers. For example, the cube of β1β is 1. Now, add the next pair of consecutive odd numbers to find the next cube. Here, 3+5= 8 which is the cube of β2β. Similarly, to find the cube of β3β, add the next set of consecutive odd numbers 7+9+11= 27.
- Difference of Cubes of Two consecutive positive integers
The difference between the two consecutive positive integers can give a cubic number. For example, The difference between 23 β 13 = 7. This represents 23 = 8. Similarly, the difference between 33 β 23 = 19 which represents 33 = 27.
- Triangular Number Pattern
Similar to square numbers, we can also find cubic numbers by adding the triangular numbers. For example, the cube number is β23β, which is the addition of the next triangular numbers i.e. (1+2=3).
Chart of Squares and Cubes 1 to 100
This chart will help students to learn the squares and cubes from 1 to 100. It will help them to solve problems of various mathematical topics such as algebra, geometry and arithmetic. Also squares and cubes are part of the number theory that will help them to deeply understand the integers.
Squares from 1 to 100
The following table shows the squares from 1 to 100:
Number | Square | Number | Square | Number | Square | Number | Square | Number | Square |
---|---|---|---|---|---|---|---|---|---|
1 | 1 | 21 | 441 | 41 | 1681 | 61 | 3721 | 81 | 6561 |
2 | 4 | 22 | 484 | 42 | 1764 | 62 | 3844 | 82 | 6724 |
3 | 9 | 23 | 529 | 43 | 1849 | 63 | 3969 | 83 | 6889 |
4 | 16 | 24 | 576 | 44 | 1936 | 64 | 4096 | 84 | 7056 |
5 | 25 | 25 | 625 | 45 | 2025 | 65 | 4225 | 85 | 7225 |
6 | 36 | 26 | 676 | 46 | 2116 | 66 | 4356 | 86 | 7396 |
7 | 49 | 27 | 729 | 47 | 2209 | 67 | 4489 | 87 | 7569 |
8 | 64 | 28 | 784 | 48 | 2304 | 68 | 4624 | 88 | 7744 |
9 | 81 | 29 | 841 | 49 | 2401 | 69 | 4761 | 89 | 7921 |
10 | 100 | 30 | 900 | 50 | 2500 | 70 | 4900 | 90 | 8100 |
11 | 121 | 31 | 961 | 51 | 2601 | 71 | 5041 | 91 | 8281 |
12 | 144 | 32 | 1024 | 52 | 2704 | 72 | 5184 | 92 | 8464 |
13 | 169 | 33 | 1089 | 53 | 2809 | 73 | 5329 | 93 | 8649 |
14 | 196 | 34 | 1156 | 54 | 2916 | 74 | 5476 | 94 | 8836 |
15 | 225 | 35 | 1225 | 55 | 3025 | 75 | 5625 | 95 | 9025 |
16 | 256 | 36 | 1296 | 56 | 3136 | 76 | 5776 | 96 | 9216 |
17 | 289 | 37 | 1369 | 57 | 3249 | 77 | 5929 | 97 | 9409 |
18 | 324 | 38 | 1444 | 58 | 3364 | 78 | 6084 | 98 | 9604 |
19 | 361 | 39 | 1521 | 59 | 3481 | 79 | 6241 | 99 | 9801 |
20 | 400 | 40 | 1600 | 60 | 3600 | 80 | 6400 | 100 | 10000 |
Cubes 1 to 100
The following table shows the cubes from 1 to 100:
Number | Cube | Number | Cube | Number | Cube | Number | Cube | Number | Cube |
---|---|---|---|---|---|---|---|---|---|
1 | 1 | 21 | 9261 | 41 | 68921 | 61 | 226981 | 81 | 531441 |
2 | 8 | 22 | 10648 | 42 | 74088 | 62 | 238328 | 82 | 551368 |
3 | 27 | 23 | 12167 | 43 | 79507 | 63 | 250047 | 83 | 571787 |
4 | 64 | 24 | 13824 | 44 | 85184 | 64 | 262144 | 84 | 592704 |
5 | 125 | 25 | 15625 | 45 | 91125 | 65 | 274625 | 85 | 614125 |
6 | 216 | 26 | 17576 | 46 | 97336 | 66 | 287496 | 86 | 636056 |
7 | 343 | 27 | 19683 | 47 | 103823 | 67 | 300763 | 87 | 658503 |
8 | 512 | 28 | 21952 | 48 | 110592 | 68 | 314432 | 88 | 681472 |
9 | 729 | 29 | 24389 | 49 | 117649 | 69 | 328509 | 89 | 704969 |
10 | 1000 | 30 | 27000 | 50 | 125000 | 70 | 343000 | 90 | 729000 |
11 | 1331 | 31 | 29791 | 51 | 132651 | 71 | 357911 | 91 | 753571 |
12 | 1728 | 32 | 32768 | 52 | 140608 | 72 | 373248 | 92 | 778688 |
13 | 2197 | 33 | 35937 | 53 | 148877 | 73 | 389017 | 93 | 804357 |
14 | 2744 | 34 | 39304 | 54 | 157464 | 74 | 405224 | 94 | 830584 |
15 | 3375 | 35 | 42875 | 55 | 166375 | 75 | 421875 | 95 | 857375 |
16 | 4096 | 36 | 46656 | 56 | 175616 | 76 | 438976 | 96 | 884736 |
17 | 4913 | 37 | 50653 | 57 | 185193 | 77 | 456533 | 97 | 912673 |
18 | 5832 | 38 | 54872 | 58 | 195112 | 78 | 474552 | 98 | 941192 |
19 | 6859 | 39 | 59319 | 59 | 205379 | 79 | 493039 | 99 | 970299 |
20 | 8000 | 40 | 64000 | 60 | 216000 | 80 | 512000 | 100 | 1000000 |
Also, Check
Solved Examples on Squares and Cubes
Here are some solved examples below:
Example 1: Find the square of number 28.
Solution:
Given number: 28
To find itβs square, multiply it twice:
Square of 28= 28β¨―28= 784
The final answer is 784.
Example 2: A square park is being constructed. The length of one side is 45 m. Find the area of the square park
Solution:
Given length: 45 m
Area of park = side2
β Area of park = 452
β Area of park = 45β¨―45
β Area of park = 2025 m2
The area of square park is 2025 m2.
Example 3: Find the square root of 144.
Solution:
Given that: 144
Square root of 144 = β144
β Square root of 144 = 12, [because 12β¨―12= 144]
Example 4: Determine a cube of 7.
Solution:
Given number: 7
cube of β7β = 73
β cube of β7β = 7β¨―7β¨―7
β cube of β7β = 343
The cube of β7β is β343β.
Example 5: Calculate the volume of a cube when its one edge is given 2 units
Solution:
Given edge: 2 units
Volume of cube = edge3
β Volume of cube = 23
β Volume of cube = 2β¨―2β¨―2
Volume of cube = 8 unit3
Practice Questions on Squares and Cubes
Here are some practice questions involving squares and cubes:
Question 1. Calculate the squares of the following numbers:
- (a) 23
- (b) 44
- (c) 65
Question 2. Find the cube root of the following integers:
- (a) 64
- (b) 512
- (c) 1331
Question 3. Choose the correct perfect square:
- (a) 81
- (b) 125
- (c) 8
Question 4. Choose the correct perfect cube:
- (a) 8
- (b) 9
- (c) 100
Question 5. A cube whose edge is 8 units. Find the volume of the cube?
Squares and Cubes β FAQs
What is Square Number?
A number which we multiply two times by itself, it becomes a square number.
How can I identify the square numbers?
In some cases, we can identify the squares through the last digit. If it ends with 1,4, 5, 6 or 9, it might be a square number. Also the difference between consecutive square numbers form an A.P.
How can I find a cube of any integers?
Multiply it three times to find the cube number. Using a calculator can also help or you can memorize a cube of numbers to a certain limit.
What is Square Root?
Square root is a reverse process of finding a square number. For example, square root of β169 is 13 because 13β¨―13= 169.
What are Cube Numbers?
Cube numbers can be found by multiplying a number thrice by itself.
What will be the Square of Integer 21?
The square of integer 21 will be 441.
Is there any pattern to find square and cube numbers?
Yes, there are many patterns to find any square number and cube number. We can identify these through the difference between consecutive squares and consecutive cubes.
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