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Chapter Analysis
Intermediate8 pages • EnglishQuick Summary
The chapter on Cubes and Cube Roots introduces students to the concept of cubes, which are numbers obtained when multiplying a number by itself three times. It explains perfect cubes through prime factorization and explores cube roots as the inverse operation of cubing. The chapter is rich with examples, exercises on finding cube roots, and real-world applications like the Hardy-Ramanujan number, emphasizing patterns related to cubes.
Key Topics
- •Understanding and identifying perfect cubes
- •Finding cube roots using prime factorization
- •Hardy-Ramanujan number and its significance
- •Patterns in cubes of even and odd numbers
- •Real-life applications and significance of cubes
- •Operations involving cubes and cube roots
Learning Objectives
- ✓Identify and determine perfect cubes through factorization
- ✓Calculate cube roots using the method of prime factorization
- ✓Explore and understand mathematical patterns related to cubes
- ✓Apply knowledge of cubes and cube roots to solve practical problems
- ✓Analyze the historical and mathematical significance of special numbers like the Hardy-Ramanujan number
Questions in Chapter
Which of the following numbers are not perfect cubes? (i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656
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Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube. (i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100
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Find the cube root of each of the following numbers by prime factorisation method. (i) 64 (ii) 512 (iii) 10648 (iv) 27000 (v) 15625 (vi) 13824 (vii) 110592 (viii) 46656 (ix) 175616 (x) 91125
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State true or false. (i) Cube of any odd number is even. (ii) A perfect cube does not end with two zeros. (iii) If square of a number ends with 5, then its cube ends with 25. (iv) There is no perfect cube which ends with 8. (v) The cube of a two digit number may be a three digit number. (vi) The cube of a two digit number may have seven or more digits. (vii) The cube of a single digit number may be a single digit number.
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Additional Practice Questions
What is the cube of 11?
easyAnswer: The cube of 11 is 11 x 11 x 11 = 1331.
If the cube of a number is 729, what is the number?
mediumAnswer: The number is 9, because 9 x 9 x 9 = 729.
Explain why 128 is not a perfect cube.
mediumAnswer: The prime factorization of 128 is 2 x 2 x 2 x 2 x 2 x 2 x 2, which cannot be grouped into triples of identical factors. Thus, 128 is not a perfect cube.
What are the cube roots of 16000 and describe the process?
hardAnswer: The cube root of 16000 is 25. This can be found by factorizing 16000 into prime factors: 16000 = 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5, then grouping in triples gives (2 x 5)³, resulting in a cube root of 25.
Provide an example of a number that is both a perfect square and a perfect cube.
mediumAnswer: One such example is 64, as 64 is 8 cubed (8 x 8 x 8) and 8 squared (8 x 8).
If the volume of a cube is 343 cm³, what is the length of its side?
easyAnswer: The length of the side is 7 cm because the cube root of 343 is 7.
What is the significance of the number 1729 in relation to cubes?
mediumAnswer: 1729 is known as the Hardy-Ramanujan number, as it is the smallest number expressible as the sum of two cubes in two different ways (10³ + 9³ and 12³ + 1³).
Describe the pattern observed in the cubes of even and odd numbers.
easyAnswer: Cubes of even numbers are always even, and cubes of odd numbers are always odd.
Demonstrate the cube root finding process using prime factorization for 27000.
hardAnswer: The prime factorization of 27000 is 2 x 3 x 3 x 3 x 5 x 5 x 5 x 2, organizing them into triples (3 x 5 x 2) gives the cube root as 30.
Identify the errors in finding cubes or cube roots in the following calculations: 729 = 2 x 2 x 2, 1000 = 3 x 3 x 3.
mediumAnswer: The number 729 is incorrect because the factorization should be 3 x 3 x 3 x 3 x 3 x 3, hence it is a perfect cube. Similarly, the factorization for 1000 should be 2 x 5 x 2 x 5 x 2 x 5 which means it is a perfect cube.