Questions and Answers
What is the cube root of 729?
The cube of an even number is always an even number.
True
What is the cube of 5?
125
The cube root of 64 is ______.
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Match the following numbers to their cube roots:
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Which statement is true regarding cubes of odd numbers?
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The sum of the first few consecutive odd numbers is always a perfect square.
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What is the cube of 4?
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The cube root of 1000 is ______.
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What is the cube root of 3375?
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A perfect cube can end with two zeros.
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What is the cube root of 8000?
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The cube root of 74088 is _____ .
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Match the following numbers with their cube roots:
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Which of the following statements is true?
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The cube of a two-digit number can have seven or more digits.
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What is the property of cubes regarding the sum of consecutive odd numbers?
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What is the prime factorisation of 216?
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A number is a perfect cube if each prime factor appears three times in its prime factorisation.
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Is the number 729 a perfect cube? Explain why.
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The prime factorisation of 500 is 2 × 2 × ___ × ___ × ___.
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Match the numbers with their properties regarding perfect cubes:
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Which number is not a perfect cube?
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All odd numbers are perfect cubes.
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The sum of the first three consecutive odd numbers is ___.
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Study Notes
Cube Roots and Prime Factorization
- The cube root symbol (³) indicates the cube root of a number.
- Examples demonstrate cube roots and their evaluations:
- ( 13^3 = 1 ) implies ( 1 = 1 )
- ( 23^3 = 8 ) implies ( 8 = 2 ) indicating that ( 2^3 = 8 )
- ( 33^3 = 27 ) implies ( 27 = 3 )
- Continuing trend with numbers up to ( 10^3 = 1000 )
Finding Cube Roots via Prime Factorization
- Prime factorization method allows calculation of cube roots:
- Example: For 3375, ( 3375 = 3^3 \times 5^3 = (3 \times 5)^3 )
- Cube root of 3375 is ( 15 )
- For 74088, ( 74088 = 2^3 \times 3^3 \times 7^3 = (2 \times 3 \times 7)^3 ) with cube root ( 42 )
- Example using ( 8000 ): ( 8000 = 2^6 \times 5^3 ) yields cube root ( 20 )
Verification of Perfect Cubes
- Perfect cubes can be identified by unique prime factors appearing in triplets:
- Example: 13824 can be factored as ( 2^3 \times 3^3 ) indicating cube root ( 24 )
- Recognize possible perfect cube numbers based on prime factorizations:
- Numbers analyzed include 4, 6, 15, and 12 exhibiting grouped factorization leading to cubes.
True or False Statements
- Evaluate statements regarding properties of cubes:
- Cube of any odd number is not even (False).
- Perfect cubes cannot end with two zeros (True).
- If a number’s square ends with 5, its cube ends with 25 (True).
- No perfect cube ends with 8 (False).
- Cube of two-digit numbers can be a three-digit number (True).
- A single-digit number's cube may also be a single-digit number (True).
Hardy-Ramanujan Numbers
- Unique numbers categorized as Hardy-Ramanujan Numbers include 1729, 4104, and 13832.
- Each prime factor in the cube's factorization appears three times, confirming structural properties of perfect cubes.
Checking for Perfect Cubes
- Determine if a number is a perfect cube by checking if each prime factor appears exactly three times.
- Example: 216's prime factorization ( 2^3 \times 3^3 ) confirms it is a perfect cube ( (2 \times 3)^3 ).
- In contrast, a number like 500, represented as ( 2^2 \times 5^3 ), is not a perfect cube due to insufficient quantity of 2's.
These notes encapsulate the key concepts and example applications of cube roots through prime factorization, alongside determining the properties and classifications of numbers in relation to perfect cubes.
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Description
Test your knowledge on calculating cube roots and understanding prime factorization. This quiz covers methods to find cube roots through prime factorization and verification of perfect cubes. Challenge yourself with examples and practice your skills!
