# 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:

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 = x1/3

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:

## 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.

## 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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