**The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.**

**A cube number is the result of multiplying a number by itself twice. For example 2**

^{3}= 2 × 2 × 2 = 8.

- When any whole number is multiplied by itself twice a cube number is formed.
- For example, 2 cubed means 2 × 2 × 2 which equals 8.
- Therefore we say that 8 is a cube number.
- ‘Cubing’ a number means to multiply a number by itself twice.
- We write a small 3 above the number to tell us to cube it like so: 2
^{3}. - Cube numbers are called this because cubing a side length of a cube gives us the volume of a cube.
- If a cube has a side length that is a whole number, its volume will be a cube number.

- To cube a number, multiply it by itself twice.
- 2 cubed is written as 2
^{3}. - 2 × 2 × 2 = 8 and so, 2 cubed is 8.
- 3 cubed is written as 3
^{3}. - 3 × 3 × 3 = 27 and so, 3 cubed is 27.
- 4 cubed is written as 4
^{3}. - 4 × 4 × 4 = 64 and so, 4 cubed is 64.

#### Cubed Numbers Video Lesson

#### Cube Numbers Activity

#### Cube Numbers Worksheets and Answers

# Cube Numbers

## What are Cube Numbers?

**A cube number is formed when any whole number is multiplied by itself twice. The symbol for cubing a number is ^{3}. For example, 2 cubed or 2^{3} means 2 × 2 × 2 which equals 8. Therefore 8 is a cube number. Cube numbers are called this because the volume of a cube is found by cubing its side length.**

Below shows a cube with side lengths of 2 cm.

The volume of the cube is found by multiplying a side length by itself twice.

2 × 2 × 2 = 8 and so the volume of the cube is 8 cm^{3}.

This means that the overall cube is made up of 8 smaller 1 cm^{3} cubes.

8 is a cube number because it is formed by multiplying a whole number by itself twice.

2 × 2 × 2 = 8

We can write this more simply as 2^{3} = 8.

The number is written to the power of 3, which is a shorter way of writing the full multiplication.

## How to Find a Cube Number

**To find a cube number, multiply any whole number by itself and then by itself again. The easiest way to do this is to do each multiplication separately. For example, 3 cubed is 3 × 3 × 3. The first multiplication is 3 × 3 = 9 and then the second multiplication is 9 × 3 = 27. Therefore the 3rd cube number is 27.**

Here is another example of calculating 4 cubed.

4^{3} means 4 × 4 × 4.

The first step is to multiply 4 × 4 to get 16.

The next step is to multiply 16 by 4 again.

16 × 4 = 64.

64 is the 4th cube number.

## List of Cube Numbers

**The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.**

Here is a complete list of the first 100 cube numbers:

Number | Calculation | Cube Number |
---|---|---|

1^{3} | 1 × 1 × 1 = | 1 |

2^{3} | 2 × 2 × 2 = | 8 |

3^{3} | 3 × 3 × 3 = | 27 |

4^{3} | 4 × 4 × 4 = | 64 |

5^{3} | 5 × 5 × 5 = | 125 |

6^{3} | 6 × 6 × 6 = | 216 |

7^{3} | 7 × 7 × 7 = | 343 |

8^{3} | 8 × 8 × 8 = | 512 |

9^{3} | 9 × 9 × 9 = | 729 |

10^{3} | 10 × 10 × 10 = | 1000 |

11^{3} | 11 × 11 × 11 = | 1331 |

12^{3} | 12 × 12 × 12 = | 1728 |

13^{3} | 13 × 13 × 13 = | 2197 |

14^{3} | 14 × 14 × 14 = | 2744 |

15^{3} | 15 × 15 × 15 = | 3375 |

16^{3} | 16 × 16 × 16 = | 4096 |

17^{3} | 17 × 17 × 17 = | 4913 |

18^{3} | 18 × 18 × 18 = | 5832 |

19^{3} | 19 × 19 × 19 = | 6859 |

20^{3} | 20 × 20 × 20 = | 8000 |

21^{3} | 21 × 21 × 21 = | 9261 |

22^{3} | 22 × 22 × 22 = | 10648 |

23^{3} | 23 × 23 × 23 = | 12167 |

24^{3} | 24 × 24 × 24 = | 13824 |

25^{3} | 25 × 25 × 25 = | 15625 |

26^{3} | 26 × 26 × 26 = | 17576 |

27^{3} | 27 × 27 × 27 = | 19683 |

28^{3} | 28 × 28 × 28 = | 21952 |

29^{3} | 29 × 29 × 29 = | 24389 |

30^{3} | 30 × 30 × 30 = | 27000 |

31^{3} | 31 × 31 × 31 = | 29791 |

32^{3} | 32 × 32 × 32 = | 32768 |

33^{3} | 33 × 33 × 33 = | 35937 |

34^{3} | 34 × 34 × 34 = | 39304 |

35^{3} | 35 × 35 × 35 = | 42875 |

36^{3} | 36 × 36 × 36 = | 46656 |

37^{3} | 37 × 37 × 37 = | 50653 |

38^{3} | 38 × 38 × 38 = | 54872 |

39^{3} | 39 × 39 × 39 = | 59319 |

40^{3} | 40 × 40 × 40 = | 64000 |

41^{3} | 41 × 41 × 41 = | 68921 |

42^{3} | 42 × 42 × 42 = | 74088 |

43^{3} | 43 × 43 × 43 = | 79507 |

44^{3} | 44 × 44 × 44 = | 85184 |

45^{3} | 45 × 45 × 45 = | 91125 |

46^{3} | 46 × 46 × 46 = | 97336 |

47^{3} | 47 × 47 × 47 = | 103823 |

48^{3} | 48 × 48 × 48 = | 110592 |

49^{3} | 49 × 49 × 49 = | 117649 |

50^{3} | 50 × 50 × 50 = | 125000 |

51^{3} | 51 × 51 × 51 = | 132651 |

52^{3} | 52 × 52 × 52 = | 140608 |

53^{3} | 53 × 53 × 53 = | 148877 |

54^{3} | 54 × 54 × 54 = | 157464 |

55^{3} | 55 × 55 × 55 = | 166375 |

56^{3} | 56 × 56 × 56 = | 175616 |

57^{3} | 57 × 57 × 57 = | 185193 |

58^{3} | 58 × 58 × 58 = | 195112 |

59^{3} | 59 × 59 × 59 = | 205379 |

60^{3} | 60 × 60 × 60 = | 216000 |

61^{3} | 61 × 61 × 61 = | 226981 |

62^{3} | 62 × 62 × 62 = | 238328 |

63^{3} | 63 × 63 × 63 = | 250047 |

64^{3} | 64 × 64 × 64 = | 262144 |

65^{3} | 65 × 65 × 65 = | 274625 |

66^{3} | 66 × 66 × 66 = | 287496 |

67^{3} | 67 × 67 × 67 = | 300763 |

68^{3} | 68 × 68 × 68 = | 314432 |

69^{3} | 69 × 69 × 69 = | 328509 |

70^{3} | 70 × 70 × 70 = | 343000 |

71^{3} | 71 × 71 × 71 = | 357911 |

72^{3} | 72 × 72 × 72 = | 373248 |

73^{3} | 73 × 73 × 73 = | 389017 |

74^{3} | 74 × 74 × 74 = | 405224 |

75^{3} | 75 × 75 × 75 = | 421875 |

76^{3} | 76 × 76 × 76 = | 438976 |

77^{3} | 77 × 77 × 77 = | 456533 |

78^{3} | 78 × 78 × 78 = | 474552 |

79^{3} | 79 × 79 × 79 = | 493039 |

80^{3} | 80 × 80 × 80 = | 512000 |

81^{3} | 81 × 81 × 81 = | 531441 |

82^{3} | 82 × 82 × 82 = | 551368 |

83^{3} | 83 × 83 × 83 = | 571787 |

84^{3} | 84 × 84 × 84 = | 592704 |

85^{3} | 85 × 85 × 85 = | 614125 |

86^{3} | 86 × 86 × 86 = | 636056 |

87^{3} | 87 × 87 × 87 = | 658503 |

88^{3} | 88 × 88 × 88 = | 681472 |

89^{3} | 89 × 89 × 89 = | 704969 |

90^{3} | 90 × 90 × 90 = | 729000 |

91^{3} | 91 × 91 × 91 = | 753571 |

92^{3} | 92 × 92 × 92 = | 778688 |

93^{3} | 93 × 93 × 93 = | 804357 |

94^{3} | 94 × 94 × 94 = | 830584 |

95^{3} | 95 × 95 × 95 = | 857375 |

96^{3} | 96 × 96 × 96 = | 884736 |

97^{3} | 97 × 97 × 97 = | 912673 |

98^{3} | 98 × 98 × 98 = | 941192 |

99^{3} | 99 × 99 × 99 = | 970299 |

100^{3} | 100 × 100 × 100 = | 1000000 |

## Properties of Cube Numbers

**Cube numbers have the following properties:
**

**
**

**An even number cubed is always even.**- An odd number cubed is always odd.
- A positive number cubed is always positive.
- A negative number cubed is always negative.
- Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit.
- The sum of the cubes of the first n natural numbers is equal to the square of their sum.

Here are some examples to illustrate these properties of cube numbers.

#### An even number cubed is always even

For example, 2 is an even number. If we cube it we get 8, which is an even answer.

2 × 2 × 2 = 8

This property works because if any number is multiplied by an even number at least once, the result is even. Cubing an even number means that we must multiply by an even number.

#### An odd number cubed is always odd

For example, 3 is an odd number. If we cube it we get 27, which is an odd answer.

3 × 3 × 3 = 27

This property works because if two odd numbers are multiplied together, the result is always odd. To make an even number, at least one of the numbers being multiplied will need an even factor. However, when an odd number is cubed, we have odd × odd × odd and so, no factor of two appears in the final result.

#### A positive number cubed is always positive

For example, 10 is a positive number. If we cube it we get 1000, which is also positive.

10 × 10 × 10 = 1000

This property works because a negative number can only be made from another negative. If we only multiply positive numbers, the result must be positive.

#### A negative number cubed is always a negative

For example, (-2) × (-2) × (-2) = -8.

When we cube a number, we multiply it by itself twice. If a negative number is cubed, we have three negative numbers multiplied together.

When three negative numbers are multiplied together, the result is always negative.

(-2) × (-2) = +4 and then 4 × (-2) = -8.

#### Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit

For example:

10^{3} = 1000. Both numbers end in 0.

21^{3} = 9261. Both numbers end in 1.

4^{3} = 64. Both Numbers end in 4.

15^{3} = 3375. Both numbers end in 5.

66^{3} = 287496. Both numbers end in 6.

19^{3} = 6859. Both numbers end in 9.

#### The sum of the cubes of the first n natural numbers is equal to the square of their sum

**1 ^{3} + 2^{3} + 3^{3} + … + n^{3} = (1 + 2 + 3 + … + n)^{2}.**

For example, for an n of 5:

1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} = 225.

(1 + 2 + 3 + 4 + 5)^{2} = 225.

Cubing the consecutive numbers and then adding them up gives the same result as adding the numbers up and then squaring them.

For example, for an n of 10:

1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + 6^{3} + 7^{3} + 8^{3} + 9^{3} + 10^{3} = 3025.

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)^{2} = 3025.

## Cube Numbers That Are Also Square Numbers

**The cube numbers that are also square numbers are found by raising whole numbers to the power of 6. For example, 1 ^{6} = 1, 2^{6} = 64 and 3^{6} = 729. 1 is 1^{2} and 1^{3}, 64 is 8^{2} and 4^{3} and 729 is 27^{2} and 9^{3}.**

1^{3} = 1 = 1^{2}
2^{3} = 8
3^{3} = 27
4^{3} = 64 = 8^{2}
5^{3} = 125
6^{3} = 216
7^{3} = 343
8^{3} = 512
9^{3} = 729 = 27^{2}
10^{3} = 1000

1, 64 and 729 are cube numbers and also square numbers.

Further cube numbers that are also square numbers are found by raising any integer to the power of 6:

1^{6} = 1
2^{6} = 64
3^{6} = 729
4^{6} = 4096
5^{6} = 15625
6^{6} = 46656
7^{6} = 117649
8^{6} = 262144
9^{6} = 531441
10^{6} = 1000000

Now try our lesson on *Finding Prime Numbers to 100* where we learn how to find prime numbers.