Generally, a cube is defined as a solid figure that has all sides equal. In other words, a Cube is a solid three-dimensional figure that consists of 6 square faces, 8 vertices, and 12 edges. When six identical square face together along with their edges form a cube. Also, the three edges of a cube join at each corner to form a vertex. The cube in a **Cube and Cube Roots **chapter is the first and basic concept that will let you know about the perfect cubes or cube numbers.

In mathematical terms, a cube of a number is explained as the multiplication of a number by three times. The cube of a number represents as a superscript 3 or 3 is written a little up to the right of the numbers.

**Representation of a cube of a number:** m³, where 3 is the power of m and read as “m cubed”.

## Cube of a Number

A number is multiplied by itself 3 times to find the cube of that number.

cube of m = m × m × m

cube of m = m³

**Examples:**

(i) Find the cube of a number 3?

**Solution:**

cube of 3 = 3³

The cube of 3 = 3³ = 3 × 3 × 3

cube of 3 = 3³ = 27

The cube of 3 is 27

(ii) Find the cube of a number 2?

**Solution:**

Cube of 2 = 2³

The cube of 2 = 2³ = 2 × 2 × 2

cube of 2 = 2³ = 8

The cube of 2 is 8

(iii) Find the cube of a number 4?

**Solution:**

cube of 4 = 4³

The cube of 4 = 4³ = 4 × 4 × 4

cube of 4 = 4³ = 64

The cube of 4 is 64

(iii) Find the cube of a number 5?

**Solution:**

Cube of 5 = 5³

The cube of 5 = 5³ = 5 × 5 × 5

cube of 5 = 5³ = 125

The cube of 5 is 125

(iv) Find the cube of a number 6?

**Solution:**

cube of 6 = 6³

The cube of 6 = 6³ = 6 × 6 × 6

cube of 6 = 6³ = 216

The cube of 6 is 216

(v) Find the cube of a number 7?

**Solution:**

cube of 7 = 7³

The cube of 7 = 7³ = 7 × 7 × 7

cube of 7 = 7³ = 343

The cube of 7 is 343

### Perfect Cubes and Cube Roots

A perfect cube is defined as a number that is the cube of an integer or the cube of some natural number.

**Examples:**

Cube of 1 = 1³ = 1 × 1 × 1 = 1

Cube of 2 = 2³ = 2 × 2 × 2 = 8

The Cube of 3 = 3³ = 3 × 3 × 3 = 27

Cube of 4 = 4³ = 4 × 4 × 4 = 64

The Cube of 5 = 5³ = 5 × 5 × 5 = 125

### Cube of Negative Numbers

Cubing the number is nothing but raising the number to its 3rd power. The Cube of a negative number is always a negative number. If -m is a number, then the cube of -m is (-m)³ = -m × -m × -m

**Examples:**

(i) Find the cube of -1?

**Answer:**

cube of -1 = (-1)³ = -1 × -1 × -1 = -1

The cube of -1 is -1

(ii) Find the cube of -2?

**Answer:**

cube of -2 = (-2)³ = -2 × -2 × -2 = -8

The cube of -2 is -8

(iii) Find the cube of -3?

**Answer:**

cube of -3 = (-3)³ = -3 × -3 × -3 = -27

The cube of -3 is -27

(iv) Find the cube of -4?

**Answer:**

cube of -4 = (-4)³ = -4 × -4 × -4 = -64

The cube of -4 is -64

(v) Find the cube of -5?

**Answer:**

cube of -5 = (-5)³ = -5 × -5 × -5 = -125

The cube of -5 is -125

### Cube of a Rational Number

Finding the cube of a rational number is represents as the (a/b)³ that is also eqaul to the a/b × a/b × a/b = (a × a × a)/(b × b × b) = a³/b³

Therefore, (a/b)³ = a³/b³

**Examples:**

(i) Find the cube of (1/2)³?

**Answer:**

cube of 1/2 = (1/2)³

The cube of 1/2 = (1/2)³ = (1/2) × (1/2) × (1/2)

cube of 1/2 = 1³/2³

cube of 1/2 = (1/2)³ = 1³/2³ = (1 × 1 × 1)/(2 × 2 × 2)

The cube of 1/2 = (1/2)³ = 1³/2³ = 1/8

The cube of (1/2) is 1/8

(i) Find the cube of (-4/3)³?

**Answer:**

cube of (-4/3) = (-4/3)³

The cube of (-4/3) = (-4/3)³ = (-4/3) × (-4/3) × (-4/3)

cube of (-4/3) = (-4)³/3³

cube of (-4/3) = (-4/3)³ = (-4)³/3³ = (-4 × -4 × -4)/(3 × 3 × 3)

The cube of (-4/3) = (-4/3)³ = (-4)³/3³ = -64/27

The cube of (-4/3) is -64/27

#### Cube number Properties:

(i) The cube of even integers is always even.

(ii) The cube of odd integers is always odd.

### Perfect Cube Solved examples

**1. Is 189 a perfect cube?**

**Answer:**

Separate 189 into different prime factors

The prime factors for 189 are 3, 3, 3, 7

189 = 3 × 3 × 3 × 7

189 = 3 × 7

Divide 189 with 7 to make it a perfect cube.

So, 189 is not a perfect cube.

**2. Find the number 216 is a perfect cube?**

**Answer:**

Firstly, find the prime factors of 216

The prime factors of 216 are 2, 2, 2, 3, 3, 3

So, 216 = 2 × 2 × 2 × 3 × 3 × 3

216 = (2 × 3) × (2 × 3) × (2 × 3)

216 = 6 × 6 × 6

The 216 = 6³ = cube of 6

216 is a perfect cube as it is the cube of 6.

216 is a perfect cube

**3. Find the smallest number that makes the 3087 a perfect cube?**

**Answer:**

To know the smallest number that makes the 3087 a perfect cube, first, we need to find the prime factors of 3087.

The 3087 prime factors are 3, 3, 7, 7, 7

3087 = 3 × 3 × 7 × 7 × 7

If the product of prime factors is multiplied by the number 3, then 3087 becomes a perfect cube.

The required number is 3

**4. Which number needs to divide from 392 to make it a perfect cube?**

**Answer:**

To find the number that makes 392 a perfect cube, we need to find the prime factors of 392

The prime factors of 392 are 2, 2, 2, 7, 7

392 = 2 × 2 × 2 × 7 × 7

The 392 will becomes a perfect cube if 7 × 7 is divided from the product of prime factors.

The required number is 7 × 7

**5. Find the cube of each of the following?**

(i) 8 (ii) (2/5) (iii) 0.2 (iv) 2 3/4 (v) -5

**Solutions:**

(i) 8

cube of 8 = 8³

The cube of 8 = 8³ = 8 × 8 × 8

cube of 8 = 8³ = 512

The cube of 8 is 512

(i)(2/5)

cube of (2/5) = (2/5)³

The cube of (2/5) = (2/5)³ = (2/5) × (2/5) × (2/5)

cube of (2/5) = (2/5)³ = 2³/5³ = (2 × 2 × 2)/(5 × 5 × 5) = 8/125

The cube of (2/5) is 8/125

(iii) 0.2

cube of 0.2 = 0.2³

The cube of 0.2 = 0.2³ = (0.2) × 0.2 × 0.2

cube of 0.2 = (0.2)³ = 0.008

The cube of (0.2) is 0.008

(iv) 2 3/4

2 3/4 = 11/4

cube of 11/4 = (11/4)³

The cube of (11/4) = (11/4)³ = (11/4) × (11/4) × (11/4)

cube of (11/4) = (11/4)³ = (11 × 11 × 11)/(4 × 4 × 4) = 1331/64

The cube of (2 3/4) is 1331/64

(i) -5

cube of -5 = -5³

The cube of -5 = -5³ = -5 × -5 × -5

cube of -5 = -5³ = -125

The cube of (-5) is -125

The cube of a number is clearly explained along with examples and explanations.