# Finding The 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.

## 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.

$$600 = 2 \times 2\times 2 \times 3 \times 5 \times 5 = 2^{3}.3.5^{2}$$

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?

Factors of $$120 = 2 \times 2 \times 2 \times 3 \times 5 = 2^{3}. 3.5$$

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.

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.

## 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.