A perfect cube is a number that can be expressed as the product of an integer multiplied by itself twice, symbolically represented as** n = x ^{3}**, where x is an integer. Understanding perfect cubes is fundamental in mathematics, particularly in

**algebra**and number theory. This article delves into the concept of perfect cube numbers, offering a comprehensive definition, step-by-step methods for identifying them, and a detailed explanation of their properties. Additionally, you’ll find a complete list of perfect cubes, alongside solved examples to solidify your grasp of the topic. Whether you’re a student or an aspirant, this guide will enhance your understanding of perfect cubes and their applications.

Table of Contents

## Definition of Perfect Cube

A perfect cube is a number that is the result of an integer multiplied by itself twice. In other words, a number n is a perfect cube if there exists an integer x such that:

n = x^{3} |

Here, x is the cube root of n, and n is said to be a perfect cube.

**Integer Cube Root**: A perfect cube always has an integer cube root. If n is a perfect cube, then ^{3}**√x = x**^{1/3}

**Odd and Even Properties**:

- The cube of an even integer is even. For example, 2
^{3}= 8 is even. - The cube of an odd integer is odd. For example, 3
^{3 }= 27 is odd.

**The sum of Consecutive Odd Numbers**: Every perfect cube can be represented as the sum of a series of consecutive odd numbers. For instance, 3^{3 }= 27 = 7+9+11.

**Digit Pattern**: The last digit of a perfect cube has a predictable pattern depending on the last digit of its root. For example:

- If the last digit of x is 2, then the last digit of x
^{3}will be 8. - If the last digit of x is 7, then the last digit of x
^{3}will be 3.

**Non-Negative Results**: The cube of any non-negative integer is non-negative, meaning x^{3 }≥ 0 for any integer x.

**Scaling Property**: If a number n is a perfect cube, then any number of the form K^{3}×n (where k is an integer) will also be a perfect cube.

**Factorization**: The prime factorization of a perfect cube will have each prime factor raised to an exponent that is a multiple of 3. For example, 216 = 2^{3}×3^{3} is a perfect cube because both exponents (3 and 3) are multiples of 3.

**Also Read: ****Ascending Order**

## Steps to Find the Perfect Cube

To determine if a number is a perfect cube or to find the cube of a number, follow these steps:

**1. Understand the Concept of Cubing**

- The cube of a number xxx is found by multiplying the number by itself twice: x
^{3 }= x × x × x

**2. Identify the Number**

- Determine the number x whose cube you want to find or verify if a given number n is a perfect cube.

**3. Calculate the Cube**

**If given x**: Compute x^{3}by multiplying x by itself twice. For example, for x=4, calculate:

4^{3 }= 4×4×4=64

- If given n: To check if n is a perfect cube, find its cube root
^{3}√n. If^{3}√n is an integer, then n is a perfect cube. For example, to check if 27 is a perfect cube, calculate:

^{3}√27 = 3 (since 3^{3 }= 27)

Therefore, 27 is a perfect cube.

**4. Check the Integer Property**

- If calculating the cube root, ensure the result is an integer. If it is not an integer, the number is not a perfect cube.

**5. Verify Using Prime Factorization (Optional)**

- Factorize the number into its prime factors. For the number to be a perfect cube, each prime factor must appear with an exponent that is a multiple of 3. For example, consider n=216

216 = 2^{3} x 3^{3}

Since the exponents of both prime factors (2 and 3) are multiples of 3, 216 is a perfect cube.

**6. Compare with a List of Known Perfect Cubes (Optional)**

- You can also compare the number with a pre-existing list of perfect cubes (e.g., 1, 8, 27, 64, 125, etc.) to verify if it is a perfect cube.

## Properties of Perfect Cube

Here is the list of properties of the perfect cube.

**Integer Cube Root**:

A perfect cube has an integer cube root. If n is a perfect cube, then ^{3}**√27 = 3, **because 27 = 3^{3}

**Odd and Even Cubes**:

The cube of an even integer is even. For example, 4^{3 }= 64 is even.

The cube of an odd integer is odd. For example, 5^{3 }= 125 is odd.

**The sum of Consecutive Odd Numbers**:

Every perfect cube can be expressed as the sum of consecutive odd numbers. For example, 3^{3 }= 27 = 7+9+11.

**Digit Patterns**:

The last digit of a perfect cube follows specific patterns depending on the last digit of the base number. For example:

If x ends in 1, x^{3 }ends in 1.

If x ends in 2, x^{3} ends in 8.

If x ends in 7, x^{3 }ends in 3.

If x ends in 9, x^{3} ends in 9.

**Non-Negative Results**:

The cube of any non-negative integer is non-negative. Thus, x^{3 }≥ 0 for any integer x. For example, (−3)^{3 }= −27, and 4^{3 }= 64.

**Scaling Property**:

If a number n is a perfect cube, then any number of the form k^{3}×n (where k is an integer) is also a perfect cube. For example, if n=8 (which is 2^{3}), then (3^{3})×8 = 216 is also a perfect cube.

**Prime Factorization**:

The prime factorization of a perfect cube will show each prime factor raised to an exponent that is a multiple of 3. For example, the prime factorization of 216 = 2^{3}×3^{3}, indicating it is a perfect cube.

**Volume Representation**:

A perfect cube can represent the volume of a cube in geometry, where the side length of the cube is an integer. For instance, a cube with a side length of 3 has a volume of 3^{3 }= 27.

**Cube Root Functionality**:

The cube root function is the inverse of the cubing function. This property allows the calculation of the original number from its cube. If n = x^{3}, then x = ^{3}√n

**Geometric Growth**:

Cubing a number results in rapid growth compared to squaring it. For instance, 2^{2 }= 4 and 2^{3 }= 8, but 10^{2 }= 100 and 10^{3}=1000. This property is important in various fields, including volume calculations and higher-dimensional analysis.

## List of Perfect Cubes up to 1000

Here is a table of the first 50 perfect cubes:

n | n^{3} | n | n^{3} |

1 | 1 | 26 | 17576 |

2 | 8 | 27 | 19683 |

3 | 27 | 28 | 21952 |

4 | 64 | 29 | 24389 |

5 | 125 | 30 | 27000 |

6 | 216 | 31 | 29791 |

7 | 343 | 32 | 32768 |

8 | 512 | 33 | 35937 |

9 | 729 | 34 | 39304 |

10 | 1000 | 35 | 42875 |

11 | 1331 | 36 | 46656 |

12 | 1728 | 37 | 50653 |

13 | 2197 | 38 | 54872 |

14 | 2744 | 39 | 59319 |

15 | 3375 | 40 | 64000 |

16 | 4096 | 41 | 68921 |

17 | 4913 | 42 | 74088 |

18 | 5832 | 43 | 79507 |

19 | 6859 | 44 | 85184 |

20 | 8000 | 45 | 91125 |

21 | 9261 | 46 | 97336 |

22 | 10648 | 47 | 103823 |

23 | 12167 | 48 | 110592 |

24 | 13824 | 49 | 117649 |

25 | 15625 | 50 | 125000 |

## Perfect Cube Numbers Solved Example

Here are five solved examples involving perfect cube numbers:

**Example 1: Finding the Cube of a Number**

**Problem:** Find the cube of 7.

**Solution:**The cube of a number n is calculated by multiplying the number by itself twice:

7^{3 }= 7×7×7 = 49×7 = 343

**Answer:** The cube of 7 is 343.

**Example 2: Checking if a Number is a Perfect Cube**

**Problem:** Determine if 729 is a perfect cube.

**Solution:**To check if 729 is a perfect cube, find its cube root:

^{3}√729 = 9

Since 9 is an integer and 9^{3 }= 729 is a perfect cube.

**Answer:** Yes, 729 is a perfect cube.

**Example 3: Expressing a Perfect Cube as a Sum of Consecutive Odd Numbers**

**Problem:** Express 4^{3} as a sum of consecutive odd numbers.

**Solution:**First, calculate 4

^{3}:

4^{3 }= 64

Now, express 64 as a sum of consecutive odd numbers:

64 = 13+15+17+19

**Answer:** 4^{3 }= 64 can be expressed as 13+15+17+19.

**Example 4: Finding a Number Given its Cube**

**Problem:** If the cube of a number is 512, what is the number?

**Solution:**Find the cube root of 512:

^{3}√512 = 8

Since 8^{3 }= 512, the number is 8.**Answer:** The number is 8.

**Example 5: Solving a Word Problem Involving a Cube**

**Problem:** A cube-shaped box has a volume of 343 cubic units. What is the length of one side of the box?

**Solution:**The volume V of a cube is given by V = s

^{3}, where sss is the side length. Given V=343, find s:

S = ^{3}√343 = 7

**Answer:** The length of one side of the box is 7 units.

**Also Read: ****What are Composite Numbers from 1 to 100?**

## FAQs

**Is 256 a perfect cube?**

No, 256 is not a perfect cube. A perfect cube is a number that can be expressed as the cube of an integer. The cube root of 256 is not an integer.

**Is 243 a perfect cube?**

No, 243 is not a perfect cube. A perfect cube is a number that can be expressed as the product of three identical integers. The cube root of 243 is not a whole number.

**Is 72 a perfect cube?**

No, 72 is not a perfect cube. A perfect cube is a number that can be expressed as the product of three identical integers. The prime factorization of 72 contains a factor of 3 that is not in groups of three, making it impossible to form a perfect cube.

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