CBSE Notes For Class 8 Maths Chapter 6 Cubes And Cube Roots

1.0Introduction

Chapter 6 of Class 8 Maths emphasizes the elementary concepts of cube and cube roots, which the students have learned in the previous chapters. These concepts, however, apply to the understanding of the three-dimensional visions of mathematics and are widely used in the practical world. Let’s take a look at the notes from the CBSE class 8 maths, chapter 6 cubes, and cube roots. 

2.0CBSE Class 8 Maths Chapter-6 Cubes and Cube Roots - Revision Notes

Important Concepts

(a) Cube Numbers

• When you multiply a number by itself three times, you get a cube number.
• The formula, where k and an are integers, is k = a × a × a = a³.
• For instance, 3³ = 3 × 3 × 3 = 27

(b) Properties of Cube Numbers

• Number cubes that are even are always even.
• Cubes with odd numbers are always odd.
• An integer cube that is negative is always negative.
• Numbers that come from multiplying an integer by itself three times are called perfect cubes.

Some topics that you will uncover in this chapter will include:

• Overview
• Cubes
• Root of the Cube
• Prime Factorisation Method for Cube Root
• Cube Root of a cube number

Definitions

Cube: The outcome of three times multiplying an integer by itself. A number that yields the original number when cubed is known as the "cube root." The ideal cube is a value that is an integer's cube.

Key Formulas and Rules

(a) Cube Formula: a³ = a × a × a

(b) Cube Root Symbol: ∛x or x^(1/3)

(c) Perfect Cubes Pattern (First 10): 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Tips & Tricks

Identifying Cube Roots:

• Break down numbers into prime factors
• Group factors in sets of three
• Multiply one number from each group

Unit Digit Pattern in Cubes: 

Practical Methods

(a) Finding Perfect Cubes:

• Use prime factorisation
• Check if all prime factors appear in groups of three
• If not, multiply by missing factors to create a perfect cube

(b) Short-Cut for Two-digit Numbers: Example for 35³:

1. Split into tens (3) and units (5)
2. Calculate step-by-step
3. Combine results systematically
4. Solved Examples

Example 1: Q: Find the cube root of 125. Solution:

• Prime factorization: 125 = 5 × 5 × 5
• Therefore, ∛125 = 5

Example 2: Q: Is 704 a perfect cube? Solution:

• 704 = 2 × 2 × 2 × 2 × 2 × 2 × 11
• 11 appears once, not in a group of three
• Therefore, 704 is not a perfect cube
• To make it a perfect cube, divide by 11

(c) Long Division Method: Example: Finding  \sqrt[3]{1728}

Step 1: Grouping

• Group the digits from right to left in sets of 3
• 1,728 becomes 1 | 728

Step 2: Find the First Digit of the Cube Root

• Look for the largest cube less than or equal to the first group
• For 1, the largest cube is 1 (1³)
• The first digit of the cube root is 12

Step 3: Subtract and Bring Down

• Subtract 1³ = 1 from the first group
• Bring down the next group (728)
• The new dividend becomes 728

Step 4: Multiply and Subtract

• Multiply the first digit (12) by 3 times the first digit: 12 × 3 × 12 = 432
• Subtract 432 from 728
• Remainder becomes new dividend

Step 5: Repeat

• Continue the process until no remainder is left
• The final result is the cube root

In this example, 

$\sqrt[3]{1728}=12$

3.0Key Features of CBSE Class 8 Maths Notes Chapter 6 Cubes and Cube Roots

Some key features of Notes for CBSE Class 8 Maths, especially for Chapter 6: Cubes and Cube Roots, include the following:

1. Thorough explanation of cube and cube root ideas
2. Methodical approach to problem-solving with an emphasis on identifying patterns
3. Applications in the real world
4. Methods for a step-by-step solution