Chapter 6: Cubes and Cube Roots
Overview of the Chapter
Introduction to the Concept of Cubes
Definition of Cubes: A cube of a number is the number multiplied by itself three times.
Properties of Cubes
Understanding Perfect Cubes
Perfect Cubes: A number is called a perfect cube if it can be expressed as the product of three equal integers.
Cube Numbers and Their Characteristics
Odd and Even Cubes: The cube of an even number is even, and the cube of an odd number is odd.
Cubes of Numbers from 1 to 10
| Number | Cube |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
Cubes of Numbers from 11 to 20
| Number | Cube |
|---|---|
| 11 | 1331 |
| 12 | 1728 |
| 13 | 2197 |
| 14 | 2744 |
| 15 | 3375 |
| 16 | 4096 |
| 17 | 4913 |
| 18 | 5832 |
| 19 | 6859 |
| 20 | 8000 |
Patterns in Cubes
Recognizing Patterns: There are several patterns observable in the cubes of numbers, which can help in identifying cube numbers.
Hardy – Ramanujan Number (1729)
Hardy – Ramanujan Number: The number 1729 is known as the Hardy-Ramanujan number, which is the smallest number that can be expressed as the sum of two cubes in two different ways.
Adding Consecutive Odd Numbers
Property of Sum of Consecutive Odd Numbers
Adding Consecutive Odd Numbers: The sum of the first ‘n’ consecutive odd numbers is equal to the cube of ‘n’.
Finding Cube Roots
Definition and Concept of Cube Roots
Cube Root: The cube root of a number is a value that, when multiplied by itself three times, gives the original number.
Methods to Find Cube Roots
Prime Factorization Method
Prime Factorization: This method involves expressing the number as a product of prime factors and then grouping them in sets of three.
Cube Roots and Their Prime Factors
Prime Factorization: Prime factors of a number play a crucial role in determining whether a number is a perfect cube. The number must have a triplet of every prime factor to be a perfect cube.
Smallest Multiple that is a Perfect Cube
Finding the Smallest Multiple: To find the smallest number that must be multiplied to make a number a perfect cube, identify the prime factors and ensure all primes appear in triplets.
Key Terms and Concepts
Cube
Definition: The result of multiplying a number by itself three times.
Cube Root
Definition: A number that produces a given number when multiplied by itself twice.
Perfect Cube
Definition: A number that is the cube of an integer.
Prime Factorization
Definition: Expressing a number as the product of its prime factors.
Hardy – Ramanujan Number
Definition: The smallest number that can be expressed as the sum of two cubes in two different ways.
Additional Value Addition
Real-life Applications
- Volume Calculation: Understanding cubes is essential in calculating the volume of cubes in geometry and real-life objects.
- Engineering and Architecture: Cube roots are used in structural calculations to determine material strength and other parameters.
Chronology of Key Concepts
- Understanding Cubes: Introduction to cubes and their properties.
- Perfect Cubes: Identification and characteristics of perfect cubes.
- Hardy – Ramanujan Number: Understanding the significance of 1729.
- Finding Cube Roots: Methods such as prime factorization and estimation.
- Smallest Multiple: How to determine the smallest multiple to make a number a perfect cube.
- Practice Problems: Application of the concepts through example problems.
Detailed Insights and Examples
Applications of Cubes in Geometry
Insight: Cubes are fundamental in understanding the geometry of three-dimensional objects.
Prime Factorization in Finding Cube Roots
Insight: Prime factorization is a reliable method for determining if a number is a perfect cube.
FAQs – Chapter 6: Cubes and Cube Roots
A cube in mathematics is the result of multiplying a number by itself three times. For example, the cube of 2 is 2 × 2 × 2 = 8.
A perfect cube is a number that can be expressed as the product of three equal integers. For example, 27 is a perfect cube because 3 × 3 × 3 = 27.
Cube numbers have specific characteristics, such as the cube of an even number being even and the cube of an odd number being odd.
The Hardy – Ramanujan Number is 1729, which is the smallest number that can be expressed as the sum of two cubes in two different ways: 1³ + 12³ and 9³ + 10³.
The sum of the first ‘n’ consecutive odd numbers is equal to the cube of ‘n’. For example, 1 + 3 + 5 = 9 = 3².
A cube root is a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27.
To find the cube root using prime factorization, express the number as a product of prime factors and then group them in sets of three. For example, the cube root of 216 is found by 216 = 2 × 2 × 2 × 3 × 3 × 3, and the cube root is 2 × 3 = 6.
The number 1729 is significant because it is the smallest number that can be expressed as the sum of two cubes in two different ways. It is also known as the Hardy – Ramanujan Number.
One pattern is that the cube of numbers ending in 1 always ends in 1. Recognizing such patterns can help identify cube numbers.
To find the smallest multiple that makes a number a perfect cube, identify the prime factors of the number and ensure all primes appear in triplets. Multiply the number by the required prime factor to make it a perfect cube.
The cube of 3 is 27, calculated as 3 × 3 × 3.
The cube root of 64 is 4, as 4 × 4 × 4 = 64.
Yes, a negative number can be a perfect cube. For example, -27 is a perfect cube because (-3) × (-3) × (-3) = -27.
The volume of a cube is calculated using the formula V = a³, where a is the length of a side of the cube.
The cube of 5 is 125, calculated as 5 × 5 × 5.
The sum of the cubes of 2 and 3 is 8 + 27 = 35.
Cubes are used in various real-life applications, such as calculating the volume of cubic containers and in engineering and architecture for structural calculations.
Prime factorization helps determine if a number is a perfect cube by breaking down the number into its prime factors. If each prime factor appears in triplets, the number is a perfect cube.
The smallest perfect cube is 1, as 1 × 1 × 1 = 1.
The cube root of 1000 is 10, as 10 × 10 × 10 = 1000.
A cube number is a number that can be expressed as the product of three identical factors. For example, 8 is a cube number because 2 × 2 × 2 = 8.
You can estimate a cube root by finding the two closest perfect cubes between which the number lies and then approximating the value.
The cube of 8 is 512, calculated as 8 × 8 × 8.
The prime factorization of 216 is 2 × 2 × 2 × 3 × 3 × 3, or 2³ × 3³.
The sum of the cubes of 4 and 5 is 64 + 125 = 189.
MCQs on Chapter 6: Cubes and Cube Roots
1. What is the cube of 4?
2. Which of the following numbers is a perfect cube?
3. What is the cube root of 27?
4. Which number is not a perfect cube?
5. What is the smallest cube number?
6. The cube of which number is 729?
7. What is the cube root of 1000?
8. Which of the following is the cube of an odd number?
9. What is the cube root of 512?
10. What is the cube of 11?
11. Which of the following numbers is the cube of 5?
12. The cube of 7 is:
13. Which of the following is not a cube of a number?
14. What is the cube of 8?
15. What is the cube root of 343?
Explore comprehensive short notes for all chapters of Class 8 Science, from the NCERT book ‘Science,’ perfect for CBSE students in India. Click here to access the complete collection.
