Perfect Cubes - Definition, List, Chart and Examples

A perfect cube is a number that is the cube of an integer. In other words, a number n is a perfect cube if there exists an integer x such that n = x³

For example, 64 is a perfect cube because 4³ = 64, and 8 is a perfect cube because 8 = 2³.

Let's discuss the definition and list of perfect cubes of numbers along with the step wise method to find them.

Cube Root of Perfect Cubes

Cube root of a perfect cube refers to the number that when multiplied by itself twice equals a perfect cube.

For example:

• Cube root of 125 is 5 because 5×5×5 = 125.
• Cube root of 216 is 6 because 6×6×6 = 216.

Read More: Cube Roots

Steps To Find The Perfect Cube

We find perfect cube of a number by following these steps

Step 1. Start by performing the prime factorization of the number. This means breaking down the number into its prime factors (factors that are prime numbers).

Step 2. Once you have the prime factors, group them into sets of three identical factors.

Step 3.If there are any prime factors that cannot be grouped into sets of three, then the number is not a perfect cube.

Step 4.If all prime factors can be grouped into sets of three, then the number is a perfect cube. The cube root of the number is obtained by multiplying one factor from each group.

Example: Let's find out if 216 is a perfect cube.

• Prime Factorization of 216: 216 = 2×2×2×3×3×3.
• Grouping: (2×2×2) and (3×3×3).
• Since all factors are grouped in sets of three, 216 is a perfect cube.
• Cube root of 216 is 2×3 = 6 (since 6³ = 216 ).

Properties of Perfect Cubes

Let's discuss some important properties of perfect cubes.

Perfect Cubes Formula

The formula for calculating a perfect cube is:

Perfect Cube = n³

where n is Integer

For instance, if n=5, then 5³= 125, making 125 a perfect cube.

Formula for factoring perfect cubes is:

a³ + b³= (a + b) × (a² − ab + b²)

This formula is an expansion of the sum of cubes. For instance, consider 8³ + 27³ :

8³ + 27³= (8 + 27) × (8² − 8 × 27 + 27²)

Solving this equation would give the factors of the sum of the cubes of 8 and 27.

Tips and Tricks to Find Perfect Cubes

Finding perfect cubes, especially for larger numbers, can be simplified with a few handy tricks. Here are some effective methods to identify perfect cubes:

Check the Cube Root

Quick Check with Cube Root: The most straightforward way to check if a number is a perfect cube is to take its cube root. If the cube root is an integer, then the number is a perfect cube. For instance, the cube root of 27 is 3, confirming that 27 is a perfect cube.

Prime Factorization

Group Factors into Threes: Break down the number into its prime factors. If all the factors can be grouped into sets of three identical factors, the number is a perfect cube. For example, (216 = 2³ times 3³). Both 2 and 3 can be grouped into sets of three, so 216 is a perfect cube.

Use the Digital Root

The digital root of a perfect cube will always be 1, 8, or 9. The digital root is found by repeatedly summing the digits of a number until you get a single-digit number. For example, the digital root of 512 is 8 (since (5 + 1 + 2 = 8)), indicating that 512 could be a perfect cube.

Recognize Patterns in Numbers

Perfect cubes exhibit specific patterns in their last digits. For example:

1. If a number ends in 1, its cube will also end in 1.
2. If a number ends in 4, its cube ends in 4.
3. If a number ends in 7, its cube ends in 3.

Recognizing these patterns can help you quickly identify potential perfect cubes.

Sum of Consecutive Odd Numbers

Sum of Odd Numbers: A perfect cube can be expressed as the sum of consecutive odd numbers. For example:

- (1³ = 1)

- (2³ = 3 + 5 = 8)

- (3³ = 7 + 9 + 11 = 27)

The number of consecutive odd numbers needed equals the cube root of the perfect cube. This trick is useful for verifying smaller perfect cubes.

Approximation and Estimation

Estimation Method: For larger numbers, approximate the cube root by narrowing down the range. For example, if you know (10³ = 1000) and (11³ = 1331), then the cube root of a number between 1000 and 1331 will be between 10 and 11. This method helps in quickly estimating whether a number is a perfect cube.

Practice with Smaller Numbers

Memorize Small Cubes: Memorizing cubes of smaller numbers (1 to 20) can help in recognizing patterns and making quick decisions about larger numbers. For example, knowing that (12³ = 1728) can make it easier to spot related cubes.

These tricks can make the process of finding and verifying perfect cubes more manageable, especially when working with large numbers or solving mathematical problems.

List of Perfect Cubes of First 5 Numbers

Perfect Cubes from 1 to 20

The cube of natural numbers 1 to 20 is discussed in the table below,

Read More: Cube 1 to 20

Perfect Cubes of Numbers from 1 to 50

Below is the table of Perfect Cubes of Numbers from 1 to 50:

Read More:List of Squares and Cubes 1 to 100

Solved Examples on Perfect Cubes

Let's solve some example problems on the concept of perfect cubes.

Example 1. Determine if 64 is a perfect cube?

Solution:

64= 4×4×4= 4³. So, 64 is a perfect cube.

Example 2. Express 512 as a perfect cube?

Solution:

512= 8×8×8= 8³. Hence, 512 is a perfect cube.

Example 3. Determine the smallest perfect cube greater than 200?

Solution:

The cube root of 200 is approximately 6.3, and the next integer is 7. Therefore, 7³ =343 is the smallest perfect cube greater than 200.

Example 4. Find the difference between two consecutive perfect cubes that have a sum of 189?

Solution:

Let the consecutive perfect cubes be n³ and (n+1)³. According to the problem, n³ +(n+1)³=189.

Solving this equation, we find n=4. Therefore, the cubes are 4³= 64 and 5³= 125.

The difference between these cubes is 125−64= 61.

Practice Problems on Perfect Cubes

Here is a worksheet on perfect cubes for you to solve now.

Problem 1: Determine if the following numbers are perfect cubes: 64, 125, 216, 200.

Problem 2: Find the cube root of 2744 and determine if it is a perfect cube.

Problem 3: Identify the smallest perfect cube greater than 1000.

Problem 4: Show that 343 is a perfect cube by expressing it as 7ⁿ.