Perfect Cube of Numbers

A perfect cube of a number is a number that is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then x = y3. Therefore, if we take the cube root of a perfect cube, we get a natural number and not a fraction. Hence, 3√x = y. For example, 8 is a perfect cube because 3√8 = 2.  The major application of cubes can be seen in geometry while determining the volume of a cube (a 3d shape). Let us learn here the definition, list of perfect cubes, applications with examples.

What are Perfect Cubes?

Perfect cubes are the numbers that are the triple product of the same number. In other words, a perfect cube is a value that is the result of three times multiplication of a whole number to itself. A perfect cube is a number, that can be written as three times the product of the same number.

The cube root of a perfect cube always results in a whole number. But we can find cubes of any number whether it is an integer or a fraction. For example, the cube of 0.5 is:

0.5 × 0.5 × 0.5 = 0.125

Let us see the list of perfect cubes in the next section.

List of Perfect Cubes 1 to 50

Number (x) Multiplied Three times by itself Cubes (x3)
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
6 6× 6× 6 216
7 7× 7× 7 343
8 8× 8× 8 512
9 9× 9× 9 729
10 10× 10× 10 1000
11 11× 11× 11 1331
12 12× 12× 12 1728
13 13× 13× 13 2197
14 14× 14× 14 2744
15 15× 15× 15 3375
16 16× 16× 16 4096
17 17× 17× 17 4913
18 18× 18× 18 5832
19 19× 19× 19 6859
20 20× 20× 20 8000
21 21× 21× 21 9261
22 22× 22× 22 10648
23 23× 23× 23 12167
24 24× 24× 24 13824
25 25× 25× 25 15625
26 26× 26× 26 17576
27 27× 27× 27 19683
28 28× 28× 28 21952
29 29× 29× 29 24389
30 30× 30× 30 27000
31 31× 31× 31 29791
32 32× 32× 32 32768
33 33× 33× 33 35937
34 34× 34× 34 39304
35 35× 35× 35 42875
36 36× 36× 36 46656
37 37× 37× 37 50653
38 38× 38× 38 54872
39 39× 39× 39 59319
40 40× 40× 40 64000
41 41× 41× 41 68921
42 42× 42× 42 74088
43 43× 43× 43 79507
44 44× 44× 44 85184
45 45× 45× 45 91125
46 46× 46× 46 97336
47 47× 47× 47 103823
48 48× 48× 48 110592
49 49× 49× 49 117649
50 50× 50× 50 125000

How to Find a Perfect Cube?

Every number can be expressed as the product of the power of its Prime factors. If the power of all the Prime factors is in the multiple of 3, then the number is said to be a perfect cube.

1) 15 = 3 × 5

153 = 15 × 15 × 15 = 3 × 5 × 3 × 5 × 3 × 5 = 33 × 53

2) 18 = 2 × 3 × 3

183 = 18 × 18 × 18 = 2 × 3 × 3 × 2 × 3 × 3 × 2 × 3 × 3 = 23 × 33 × 33

From the above examples, we can observe that each prime factor of a number appears three times in the prime factorization of its cube.

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Solved Examples on Perfect Cubes

Question 1: Is the number 600 a perfect cube?

Let us write down, stating the prime factors of the number 600.

\(\begin{array}{l}600 = 2 \times 2\times 2 \times 3 \times 5 \times 5 = 2^{3}.3.5^{2} \end{array} \)

The number 600 is not a perfect cube as all the prime factors is not a multiple of three.

Question 2: What should be multiplied to the number 120 to make it a perfect cube?

\(\begin{array}{l} Factors\ of\ 120 = 2 \times 2 \times 2 \times 3 \times 5  = 2^{3}. 3.5\end{array} \)

The factor 2 occurs thrice, 3 and 5 occurs only once. So in order to make it a perfect cube we have to multiply 120 with a square of 3 and 5.

Therefore, 120 × 3 × 3 × 5 × 5 =27000

The number 27000 is the required perfect cube.

Application of Perfect Cubes

A cube is a three-dimensional figure which has all the edges of equal length. The volume of a cube is given by the product of its dimensions. For example, for a cube of edge length ‘a’ units, the volume of the cube is given by “a × a × a”which is equal to a3 cubic unit. Similarly, we can calculate the perfect cube of various numbers.

Perfect Cube

Illustration: Determine the volume of a cube with an edge length of 7 cm.

Solution: The side of the cube is 7 cm.

Volume = side × side × side = 7 × 7 × 7 = 343 cm3.

The numbers such as 1, 8, 27, 64, etc. are known as perfect cubes or cube numbers.

When a cube is broken or cut, total volume still remains the same. Hence, we can observe that to construct a cube of edge length ‘2a’ with cubes of edge length ‘a’ we need to have 8 cubes of edge length a.

This article explains perfect cubes and the method of their calculation. To learn more about other topics download BYJU’S – The Learning App and watch interactive videos. Also, take free tests to practice for exams.

Frequently Asked Questions on Perfect Cubes

What is a perfect cube?

When a natural number is multiplied three times to itself, then the resulting value is called perfect cube.

What is the perfect cube of 100?

The prefect cube of 100 is 100 x 100 x 100 = 1000000

What are first ten perfect cube numbers?

The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.

Which are the perfect cubes among these: 27, 9, 400?

27 is the perfect cube. Since 3 x 3 x 3 = 33 = 27
9 and 400 are the perfect squares. 3 x 3 = 9 and 20 x 20 = 400

What is the cube 30?

The cube of 30 is 27000.
Test your Knowledge on Perfect Cube of Numbers

2 Comments

  1. I hate math but thank you for this so I can learn

  2. very good efforts , it really help in doing my work

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