Perfect Cubes – Definition, List, Chart and Examples

Perfect cubes are numbers that result from multiplying an integer by itself twice. A number is said to be a perfect cube if it can be decomposed into a product of the same number thrice.

List of Perfect Cubes

Let’s discuss the definition and list of perfect cubes of numbers along with the stepwise method to find them.

Perfect Cubes Definition

A perfect cube refers to a number that results from the product of a number multiplied by itself twice.

For a number to qualify as a perfect cube, it must be possible to find a number that when multiplied by itself twice, gives the same number.

For example, 8 is a perfect cube because 8 = 2 × 2 × 2 = 2³.

List of Perfect Cubes

Number	Multiplied three times	Cube
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

Perfect Cubes from 1 to 100

To identify the perfect cubes between 1 and 100, we search for numbers that are perfect cubes within this range. In this case, the perfect cubes within this range are 1, 8, 27, and 64.

1³ = 1

2³ = 8

3³ = 27

4³ = 64

Therefore, within the range of 1 to 100, there are precisely three perfect cubes, namely 8, 27, and 64.

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.

How 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.

Finding out if 216 is a Perfect Cube.

• Prime Factorization of 216: 216 = 2×2×2×3×3×3 = 216 = 2×2×2×3×3×3.
• Grouping: (2×2×2)(2×2×2) and (3×3×3)(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 ).

Perfect Cubes Formula

The formula for calculating a perfect cube is:

Perfect Cube = n³

where n is Integer

Finding the Perfect Cube

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.

Perfect Cubes of Numbers from 1 to 50

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

Number	Cube
1	1³ = 1
2	2³ = 8
3	3³ = 27
4	4³ = 64
5	5³ = 125
6	6³ = 216
7	7³ = 343
8	8³ = 512
9	9³ = 729
10	10³ = 1000
11	11³ = 1331
12	12³ = 1728
13	13³ = 2197
14	14³ = 2744
15	15³ = 3375
16	16³ = 4096
17	17³ = 4913
18	18³ = 5832
19	19³ = 6859
20	20³ = 8000
21	21³ = 9261
22	22³ = 10648
23	23³ = 12167
24	24³ = 13824
25	25³ = 15625
26	26³ = 17576
27	27³ = 19683
28	28³ = 21952
29	29³ = 24389
30	30³ = 27000
31	31³ = 29791
32	32³ = 32768
33	33³ = 35937
34	34³ = 39304
35	35³ = 42875
36	36³ = 46656
37	37³ = 50653
38	38³ = 54872
39	39³ = 59319
40	40³ = 64000
41	41³ = 68921
42	42³ = 74088
43	43³ = 79507
44	44³ = 85184
45	45³ = 91125
46	46³ = 97336
47	47³ = 103823
48	48³ = 110592
49	49³ = 117649
50	50³ = 125000

Read More:

• List of Squares and Cubes 1 to 100

Properties of Perfect Cubes

Let’s discuss some important properties of perfect cubes.

Property	Description
Result of Cubing an Integer	A perfect cube is the result of multiplying an integer by itself twice.
Negative Numbers Can Form Perfect Cubes	Negative integers can form perfect cubes, e.g., (−3)3 = −27
Unique Cubes for Each Integer	Each integer has a unique cube. No two different integers have the same cube.
Zero is a Perfect Cube	Zero is considered a perfect cube because 03 = 0.
Digit Pattern	Units digit of a perfect cube can only be 0, 1, 4, 5, 6, or 9.
Factors	If a number is a perfect cube, then its prime factors are grouped in triples.
Roots	Cube root of a perfect cube is an integer.
Geometric Representation	In geometry, a perfect cube represents a three-dimensional space with equal sides.

Solved Examples on Perfect Cubes

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

1. Determine if 64 is a perfect cube?

Solution:

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

2. Express 512 as a perfect cube?

Solution:

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

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.

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

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.

Perfect Cube Numbers- FAQs

1. What are perfect cubes?

A perfect cube is a number obtained by multiplying an integer by itself twice, denoted as n³ where n is an integer.

2. How to Find if a Number is Perfect Cube or Not?

To verify if a number is a perfect cube, calculate its cube root. If the result is an integer, the number is a perfect cube.

3. Can Perfect Cube Numbers Be Negative?

Yes, perfect cube numbers can be negative. A perfect cube is a number that can be expressed as the cube of an integer, and this definition applies to negative integers as well. When a negative integer is cubed, the result is a negative perfect cube. For example, (−2)³= −8, so -8 is a perfect cube.

4. How many perfect cubes are there from 1 to 100?

The perfect cubes within the range of 1 to 100 are 1, 8, 27, and 64, which are the cubes of 1, 2, 3, and 4, respectively.

5. Is 1 a Perfect Cube?

Yes, 1 is a perfect cube. A perfect cube is a number that can be expressed as the cube of an integer. Since 1 cubed is 1, it meets the definition of a perfect cube.

6. Is 27 a Perfect Cube?

Yes, 27 is a perfect cube. A perfect cube is a number that can be expressed as the cube of an integer. Since 3 cubed is 27, it meets the definition of a perfect cube

7. Is 343 a Perfect Cube?

Yes, 343 is a perfect cube. It is the cube of 7, as 7×7×7=343. This means that 343 is the result of multiplying the number 7 by itself three times.