CLASS 8 NCERT WORKSHEET CH-1 CUBES AND CUBE ROOTS
Worksheet: Exploring Cubes and Cube RootsInstructions: Read each question carefully and use the information from the provided text to answer. Show your work where applicable.
Part A: Understanding Cubes
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Define a "perfect cube" in your own words.
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How is the cube of any number 'n' denoted mathematically? n × n × n = __________
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Complete the table of cubes for the given natural numbers:
- 1³ = __________
- 2³ = __________
- 3³ = __________
- 4³ = __________
- 5³ = __________
- 6³ = __________
- 7³ = __________
- 8³ = __________
- 9³ = __________
- 10³ = __________
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Based on the cubes you filled in the table, what are the possible digits that a perfect cube can end with (its units place)?
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If a number contains zeros at the end, what can you say about the number of zeros its cube will have at the end? For example, can a cube end with exactly two zeros (00)? Explain.
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Explore the pattern showing the sum of consecutive odd numbers resulting in cube numbers:
- 1 = 1 = 1³
- 3 + 5 = __________ = 2³
- 7 + 9 + 11 = __________ = 3³
- 13 + 15 + 17 + 19 = __________ = __________
- 21 + 23 + 25 + 27 + 29 = __________ = __________
Part B: Finding Cube Roots
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Define "cube root" and state how it is denoted.
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How can prime factorization help in determining if a number is a perfect cube, and if so, in finding its cube root?
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Determine if the following numbers are perfect cubes using prime factorization. If they are, find their cube root.
- (i) 64
- Prime factorization of 64: ___________________________________
- Can factors be grouped into triplets? (Yes/No): __________
- Is 64 a perfect cube? __________
- If yes, ³√64 = __________
- (ii) 500
- Prime factorization of 500: ___________________________________
- Can factors be grouped into triplets? (Yes/No): __________
- Is 500 a perfect cube? __________
- If yes, ³√500 = __________
- (i) 64
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Find the cube roots of the following numbers using prime factorization:
- (i) 27000
- ³√27000 = __________
- (ii) 10648
- ³√10648 = __________
- (i) 27000
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What number will you multiply by 1323 to make it a cube number? (Show your prime factorization and reasoning)
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Without using factorization, guess the cube roots of the following numbers based on patterns you might observe:
- (i) 1331 (Hint: Consider what number's cube ends in 1)
- ³√1331 = __________
- (ii) 4913 (Hint: Consider what number's cube ends in 3)
- ³√4913 = __________
- (i) 1331 (Hint: Consider what number's cube ends in 1)
Part C: True or False Statements
State whether the following statements are true or false. Explain your reasoning for each:
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The cube of any odd number is even.
- True / False: __________
- Reasoning: _______________________________________________________________________________
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There is no perfect cube that ends with 8.
- True / False: __________
- Reasoning: _________________________________________________________________________
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The cube of a 2-digit number may be a 3-digit number.
- True / False: __________
- Reasoning: _________________________________________________________________________
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The cube of a 2-digit number may have seven or more digits.
- True / False: __________
- Reasoning: _________________________________________________________________________
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Cube numbers have an odd number of factors.
- True / False: __________
- Reasoning: _________________________________________________________________________
Part D: Challenge and Historical Context
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Taxicab numbers are numbers that can be expressed as the sum of two cubes in two different ways. The number 1729 is the smallest taxicab number. How is it expressed as the sum of two cubes in two different ways?
- 1729 = __________ + __________
- 1729 = __________ + __________
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In ancient Indian Sanskrit works, what term was used for a solid cube and also for the product of a number with itself three times? What term was used for the cube root?
- Term for cube: __________
- Term for cube root: __________
Answer Key: Exploring Cubes and Cube Roots
Part A: Understanding Cubes
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Define a "perfect cube" in your own words. A perfect cube is a number obtained by multiplying a number by itself three times. For example, 8 is a perfect cube because 2 × 2 × 2 = 8.
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How is the cube of any number 'n' denoted mathematically? n × n × n = n³.
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Complete the table of cubes for the given natural numbers:
- 1³ = 1
- 2³ = 8
- 3³ = 27
- 4³ = 64
- 5³ = 125
- 6³ = 216
- 7³ = 343 (calculated from source 29, 17³=4913 ends in 3, 7³ ends in 3)
- 8³ = 512
- 9³ = 729
- 10³ = 1000
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Based on the cubes you filled in the table, what are the possible digits that a perfect cube can end with (its units place)? By observing the cubes in the table, the possible units digits for perfect cubes are 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. (Unlike squares, cubes can end in any digit)
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If a number contains zeros at the end, what can you say about the number of zeros its cube will have at the end? For example, can a cube end with exactly two zeros (00)? Explain. If a number contains zeros at the end, its cube will have a number of zeros that is a multiple of three. For example, 10³ = 1000 (one zero becomes three), 20³ = 8000 (one zero becomes three), 100³ = 1,000,000 (two zeros become six). No, a cube cannot end with exactly two zeros (00) because the number of zeros at the end of a cube must be a multiple of three.
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Explore the pattern showing the sum of consecutive odd numbers resulting in cube numbers:
- 1 = 1 = 1³
- 3 + 5 = 8 = 2³
- 7 + 9 + 11 = 27 = 3³
- 13 + 15 + 17 + 19 = 64 = 4³
- 21 + 23 + 25 + 27 + 29 = 125 = 5³
Part B: Finding Cube Roots
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Define "cube root" and state how it is denoted. If y = x³, then x is the cube root of y. It is denoted by the symbol ³√.
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How can prime factorization help in determining if a number is a perfect cube, and if so, in finding its cube root? Prime factorization helps by allowing you to check if the prime factors of a number can be grouped into three identical groups (triplets). If they can, the number is a perfect cube. The cube root is then the product of the prime factors within one of these identical groups. Each prime factor of a number will appear three times in the prime factorization of its cube.
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Determine if the following numbers are perfect cubes using prime factorization. If they are, find their cube root.
- (i) 64
- Prime factorization of 64: 2 × 2 × 2 × 2 × 2 × 2 (or 2⁶)
- Can factors be grouped into triplets? (Yes/No): Yes (2x2x2) x (2x2x2) or (2x2) x (2x2) x (2x2).
- Is 64 a perfect cube? Yes.
- If yes, ³√64 = 4 (since 4³=64).
- (ii) 500
- Prime factorization of 500: 2 × 2 × 5 × 5 × 5
- Can factors be grouped into triplets? (Yes/No): No. The factor 2 appears only twice, not three times.
- Is 500 a perfect cube? No.
- If yes, ³√500 = N/A
- (i) 64
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Find the cube roots of the following numbers using prime factorization:
- (i) 27000
- 27000 = 27 × 1000 = (3 × 3 × 3) × (10 × 10 × 10) = 3³ × 10³ = (3 × 10)³ = 30³
- ³√27000 = 30
- (ii) 10648
- 10648 = 2 × 5324 = 2 × 2 × 2662 = 2 × 2 × 2 × 1331 = 2³ × 11³ = (2 × 11)³ = 22³
- ³√10648 = 22
- (i) 27000
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What number will you multiply by 1323 to make it a cube number? (Show your prime factorization and reasoning) Prime factorization of 1323: 3 × 3 × 3 × 7 × 7 (or 3³ × 7²) To make it a perfect cube, each prime factor must appear in triplets. Here, 3 already appears as a triplet (3³), but 7 appears only twice (7²). Therefore, we need one more factor of 7 to complete the triplet for 7. So, you will multiply 1323 by 7.
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Without using factorization, guess the cube roots of the following numbers based on patterns you might observe:
- (i) 1331
- Hint: The number ends in 1. Looking at the cube table, numbers ending in 1 have cube roots ending in 1 (e.g., 1³=1, 11³=1331). Since 10³=1000, 1331 is likely 11³.
- ³√1331 = 11
- (ii) 4913
- Hint: The number ends in 3. Looking at the cube table, numbers ending in 3 have cube roots ending in 7 (e.g., 7³=343, 17³=4913).
- ³√4913 = 17
- (i) 1331
Part C: True or False Statements
State whether the following statements are true or false. Explain your reasoning for each:
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The cube of any odd number is even.
- True / False: False
- Reasoning: The cube of an odd number is always an odd number. For example, 1³ = 1, 3³ = 27, 5³ = 125, which are all odd.
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There is no perfect cube that ends with 8.
- True / False: False
- Reasoning: The number 8 itself is a perfect cube (2³ = 8). Also, 512 is a perfect cube (8³ = 512) and 12³ ends in 8.
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The cube of a 2-digit number may be a 3-digit number.
- True / False: False
- Reasoning: The smallest 2-digit number is 10. Its cube is 10³ = 1000, which is a 4-digit number. Any 2-digit number larger than 10 will have a cube greater than 1000.
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The cube of a 2-digit number may have seven or more digits.
- True / False: False
- Reasoning: The largest 2-digit number is 99. Its cube, 99³, is 970,299, which has 6 digits.
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Cube numbers have an odd number of factors.
- True / False: False
- Reasoning: Only square numbers have an odd number of factors. For a number to have an odd number of factors, it must have at least one factor that pairs with itself (e.g., 6x6 for 36). A number is a perfect cube if its prime factors can be split into three identical groups. A non-square cube, such as 8 (2x2x2), has factors 1, 2, 4, 8 (an even number of factors).
Part D: Challenge and Historical Context
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Taxicab numbers are numbers that can be expressed as the sum of two cubes in two different ways. The number 1729 is the smallest taxicab number. How is it expressed as the sum of two cubes in two different ways?
- 1729 = 1³ + 12³
- 1729 = 9³ + 10³
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In ancient Indian Sanskrit works, what term was used for a solid cube and also for the product of a number with itself three times? What term was used for the cube root?
- Term for cube: ghana
- Term for cube root: ghana-mula
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