NCERT Class 8 Mathematics

Chapter 6: Cubes and Cube Roots

Welcome to Chapter 6 of your Class 8 Mathematics NCERT textbook! This chapter introduces the fascinating concepts of cubes and cube roots, which are fundamental operations in mathematics with numerous practical applications.

What is a Cube?

When we multiply a number by itself three times, the result is called its cube.

For any number n, its cube = n × n × n = n³

For example, the cube of 4 is calculated as:

4 × 4 × 4 = 64, so 4³ = 64

The term "cube" comes from geometry, where a cube is a three-dimensional shape with all sides equal. The volume of a cube with side length 'a' is a³.

Perfect Cubes

Numbers that are the cube of integers are called perfect cubes. The first ten perfect cubes are:

1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125,
6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000

Notice that the cubes of even numbers are even, and the cubes of odd numbers are odd.

What is a Cube Root?

The cube root of a number is the value that, when multiplied by itself three times, gives the original number.

If a³ = b, then the cube root of b = a, written as ∛b = a

For example:

Since 5 × 5 × 5 = 125, then ∛125 = 5

The symbol for cube root is ∛, which is called the radical sign with a small 3.

Properties of Cubes and Cube Roots

Cube numbers and cube roots have several interesting properties:

The cube of a negative number is always negative: (-n)³ = -n³
The cube root of a negative number is also negative: ∛(-n) = -∛n
For any integer a, ∛(a³) = a
For any two integers a and b, ∛(a × b) = ∛a × ∛b
For any two integers a and b (where b ≠ 0), ∛(a ÷ b) = ∛a ÷ ∛b

Applications in Real Life

The concepts of cubes and cube roots have practical applications in various fields:

Volume calculations: Determining the volume of cubic containers
Engineering: Designing structures with cubic components
Physics: Working with three-dimensional measurements
Computer graphics: Working with 3D models and environments
Economics: Analyzing growth patterns that follow cubic functions

What You Will Learn

In this chapter, you will learn how to:

Identify perfect cubes among given numbers
Calculate the cube of any given number
Find the cube root of perfect cubes
Estimate the cube root of non-perfect cubes
Solve problems involving cubes and cube roots
Apply these concepts to real-life situations

Key Points to Remember

The cube of a number is the result of multiplying the number by itself three times
Perfect cubes are numbers that can be expressed as the cube of an integer
The cube root of a number is the value that when cubed gives the original number
Cubes and cube roots are inverse operations, just like squares and square roots
Unlike square roots, cube roots can be negative, since the cube of a negative number is negative

As you progress through this chapter, you'll discover patterns and relationships that make working with cubes and cube roots both interesting and useful. Keep your mind open to recognizing these patterns, as they will help you solve problems more efficiently!