All Perfect Cube Numbers: Definition, Steps, Properties, and Solved Examples

A perfect cube is a number that can be expressed as the product of an integer multiplied by itself twice, symbolically represented as n = x³, 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.

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 = x3

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, 2³ = 8 is even.
• The cube of an odd integer is odd. For example, 3³ = 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, 3³ = 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 x³ will be 8.
• If the last digit of x is 7, then the last digit of x³ will be 3.

Non-Negative Results: The cube of any non-negative integer is non-negative, meaning x³ ≥ 0 for any integer x.

Scaling Property: If a number n is a perfect cube, then any number of the form K³×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 = 2³×3³ 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: x³ = 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 x³ by multiplying x by itself twice. For example, for x=4, calculate: 

4³ = 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 3³ = 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 = 2³ x 3³

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 = 3³

Odd and Even Cubes:

The cube of an even integer is even. For example, 4³ = 64 is even.

The cube of an odd integer is odd. For example, 5³ = 125 is odd.

The sum of Consecutive Odd Numbers:

Every perfect cube can be expressed as the sum of consecutive odd numbers. For example, 3³ = 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, x³ ends in 1.

If x ends in 2, x³ ends in 8.

If x ends in 7, x³ ends in 3.

If x ends in 9, x³ ends in 9.

Non-Negative Results:

The cube of any non-negative integer is non-negative. Thus, x³ ≥ 0 for any integer x. For example, (−3)³ = −27, and 4³ = 64.

Scaling Property:

If a number n is a perfect cube, then any number of the form k³×n (where k is an integer) is also a perfect cube. For example, if n=8 (which is 2³), then (3³)×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 = 2³×3³, 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 3³ = 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 = x³, then x = ³√n

Geometric Growth:

Cubing a number results in rapid growth compared to squaring it. For instance, 2² = 4 and 2³ = 8, but 10² = 100 and 10³=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	n3	n	n3
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:

7³ = 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 9³ = 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 4³ as a sum of consecutive odd numbers.

Solution:
First, calculate 4³:

4³ = 64

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

64 = 13+15+17+19

Answer: 4³ = 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 8³ = 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 = s³, 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?

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