A perfect cube is a number which is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then x = y^{3}. Therefore, if we take out the cube root of a perfect cube, we get a natural number and not a fraction. Hence, ^{3}√x = y. For example, 8 is a perfect cube because ^{3}√8 = 2.

## List of Perfect Cube Numbers 1 to 50

Number (x) | Multiplied Three times by itself | Cubes (x^{3}) |

1 | 1× 1× 1 | 1 |

2 | 2× 2× 2 | 8 |

3 | 3× 3× 3 | 27 |

4 | 4× 4× 4 | 64 |

5 | 5× 5× 5 | 125 |

6 | 6× 6× 6 | 216 |

7 | 7× 7× 7 | 343 |

8 | 8× 8× 8 | 512 |

9 | 9× 9× 9 | 729 |

10 | 10× 10× 10 | 1000 |

11 | 11× 11× 11 | 1331 |

12 | 12× 12× 12 | 1728 |

13 | 13× 13× 13 | 2197 |

14 | 14× 14× 14 | 2744 |

15 | 15× 15× 15 | 3375 |

16 | 16× 16× 16 | 4096 |

17 | 17× 17× 17 | 4913 |

18 | 18× 18× 18 | 5832 |

19 | 19× 19× 19 | 6859 |

20 | 20× 20× 20 | 8000 |

21 | 21× 21× 21 | 9261 |

22 | 22× 22× 22 | 10648 |

23 | 23× 23× 23 | 12167 |

24 | 24× 24× 24 | 13824 |

25 | 25× 25× 25 | 15625 |

26 | 26× 26× 26 | 17576 |

27 | 27× 27× 27 | 19683 |

28 | 28× 28× 28 | 21952 |

29 | 29× 29× 29 | 24389 |

30 | 30× 30× 30 | 27000 |

31 | 31× 31× 31 | 29791 |

32 | 32× 32× 32 | 32768 |

33 | 33× 33× 33 | 35937 |

34 | 34× 34× 34 | 39304 |

35 | 35× 35× 35 | 42875 |

36 | 36× 36× 36 | 46656 |

37 | 37× 37× 37 | 50653 |

38 | 38× 38× 38 | 54872 |

39 | 39× 39× 39 | 59319 |

40 | 40× 40× 40 | 64000 |

41 | 41× 41× 41 | 68921 |

42 | 42× 42× 42 | 74088 |

43 | 43× 43× 43 | 79507 |

44 | 44× 44× 44 | 85184 |

45 | 45× 45× 45 | 91125 |

46 | 46× 46× 46 | 97336 |

47 | 47× 47× 47 | 103823 |

48 | 48× 48× 48 | 110592 |

49 | 49× 49× 49 | 117649 |

50 | 50× 50× 50 | 125000 |

## How to Find a Perfect Cube?

Every number can be expressed as the product of the power of its Prime factors. If the power of all the Prime factors is in the multiple of 3, then the number is said to be a perfect cube.

1) 15 = 3 × 5

15^{3 }= 15 × 15 × 15 = 3 × 5 × 3 × 5 × 3 × 5 = 3^{3 }× 5^{3}

2) 18 = 2 × 3 × 3

18^{3} = 18 × 18 × 18 = 2 × 3 × 3 × 2 × 3 × 3 × 2 × 3 × 3 = 2^{3} × 3^{3 }× 3^{3}

From the above examples, we can observe that each prime factor of a number appears three times in the prime factorization of its cube.

### Solved Examples

**Question 1: Is the number 600 a perfect cube?**

Let us write down, stating the prime factors of the number 600.

\(600 = 2 \times 2\times 2 \times 3 \times 5 \times 5 = 2^{3}.3.5^{2} \)

The number 600 is not a perfect cube as all the prime factors is not a multiple of three.

**Question 2: What should be multiplied to the number 120 to make it a perfect cube?**

Factors of \( 120 = 2 \times 2 \times 2 \times 3 \times 5 = 2^{3}. 3.5\)

The factor 2 occurs thrice, 3 and 5 occurs only once. So in order to make it a perfect cube we have to multiply 120 with a square of 3 and 5.

Therefore, 120 × 3 × 3 × 5 × 5 =27000

The number 27000 is the required perfect cube.

## Application of Perfect Cube

A cube is a three-dimensional figure which has all the edges of equal length. The volume of a cube is given by the product of its dimensions. For example, for a cube of edge length ‘a’ units, the volume of the cube is given by “a × a × a”which is equal to a^{3 }cubic unit. Similarly, we can calculate the perfect cube of various numbers.

**Illustration: Determine the volume of a cube with an edge length of 7 cm.**

Solution: The side of the cube is 7 cm.

Volume = side × side × side = 7 × 7 × 7 = 343 cm^{3}.

The numbers such as 1, 8, 27, 64, etc. are known as perfect cubes or cube numbers.

When a cube is broken or cut, total volume still remains the same. Hence, we can observe that to construct a cube of edge length ‘2a’ with cubes of edge length ‘a’ we need to have 8 cubes of edge length a.

This article explains perfect cubes and the method of its calculation. To learn more about other topics download BYJU’S – The Learning App and watch interactive videos. Also, take free tests to practice for exams.

## Frequently Asked Questions – FAQs

### What is the perfect cube of 100?

### What are first ten perfect cube numbers?

### Which are the perfect cubes among these: 27, 9, 400?

^{3}= 27

9 and 400 are the perfect squares. 3 x 3 = 9 and 20 x 20 = 400

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