Cube and cube roots – meaning and examples

When a number is multiplied three times by itself, the value we get is the cube of that number. That number which is multiplied thrice is called the cube root of that value. This means that the cube and the cube root are the inverse functions of each other. In this article, we are going to discuss the cube and the cube roots, their symbol, and some illustrations.

Cube and cube roots

Meaning

On multiplying a number three times by itself, the product we get is the cube of that number. The number which is multiplied thrice is the cube root of the product.

For example:

2³ = 2 × 2 × 2 = 8. Here, 8 is the perfect cube of 2 and 2 is the cube root of 8.

We can take another example:

3³ = 3 × 3 × 3 = 27. Here, 27 is the cube of 3 and 3 is the cube root of 27.

Symbol

The algebraic expression, i.e., the symbol of the cube is x³, where x is any number. The symbol for cube root is denoted by ” ∛  ” which is also called a radical. We know that the cube and the cube root are inverse functions of each other.

For example:

p³ = 8  (Applying the cube roots on both sides)

p = ∛8

p = 8^1/3

p = (2×2×2)^1/3

Therefore, p = 2

How to find the value of cube and the cube roots

To find the value of the cube of any number, we have to multiply that number three times by itself.

For example:

5³ = 5 × 5 × 5 = 125.

To find the value of the cube root, we generally use the prime factorisation method in which we make the factors into groups of three.

For example:

The prime factors of 343 = 7 × 7 × 7 = 7³, so the cube root of 343 is 7.

Examples

Q1. Which number’s cube will we get when we multiply the cube of 5 and 2.

Solution:

The cube of 5 = 5 × 5 × 5 = 125 & the cube of 2 = 2 × 2 × 2 = 8.

The product of cubes of 5 and 2 = 125 × 8 = 1000.

The prime factors of 1000 are 2 × 2 × 2 × 5 × 5 × 5 = 10 × 10 × 10.

Which can be written as 10³.

So, we will get the cube of 10 which is 1000, when we multiply the cube of 5 and 2.

Q2. Solve 8³/4³ = a³.

Solution:

We know that the cube of 8 = 512 and the cube of 4 = 64.

So, 512/64= a³

a³ = 8

a = ∛8

a = ∛(2×2×2)

a = ∛2³

Therefore, a = 2.