Integer exponent laws

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Formula

Explanation

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Worked solution

1. The total number of cones is 64, and they are distributed equally among 14 crates. The number of cones per crate is given by 64 ÷ 14.
2. Express 64 as a power of 2: 64 = 2^6.
3. The number of crates is 14, which is not a power of 2, but the number of cones per crate must be expressed as a power of 2. Therefore, we must find 2^6 ÷ 14 and simplify using exponent laws.
4. Since 14 is not a power of 2, we cannot directly apply a^m / a^n = a^(m-n) unless we express 14 in terms of powers of 2. However, 14 = 2 × 7, so 64 ÷ 14 = 2^6 / (2 × 7) = (2^6 / 2^1) / 7.
5. Apply the exponent law a^m / a^n = a^(m-n): 2^6 / 2^1 = 2^(6-1) = 2^5.
6. Now we have 2^5 / 7 = 32 / 7. This is not a power of 2, but the problem states that the number of cones per crate is expressed as a power of 2. This suggests a contradiction unless the initial assumption about equal distribution is re-examined.
7. Re-examining the problem: 64 cones in 14 crates, each crate holds the same number of cones, and the number per crate is a power of 2. Since 64 ÷ 14 = 32/7 ≈ 4.57, and 4.57 is not a power of 2, but 4 = 2^2 and 8 = 2^3, we test if 64 ÷ 14 can be simplified to a power of 2.
8. Check if 64 ÷ 14 simplifies to a power of 2. 64 ÷ 14 = 32 ÷ 7. 32 is 2^5, 7 is not a power of 2. Therefore, 32/7 is not a power of 2. This implies that 14 crates cannot hold 64 cones with each crate having a number of cones that is a power of 2, unless the total is not 64 or the number of crates is not 14.
9. But the problem states that 64 cones are stored in 14 crates, and the number per crate is a power of 2. Therefore, the only way this is possible is if the number of cones per crate is 2^k for some integer k, and 14 × 2^k = 64.
10. Solve 14 × 2^k = 64. Divide both sides by 14: 2^k = 64 / 14 = 32 / 7. But 32/7 is not a power of 2. Therefore, no integer k satisfies this equation.
11. This indicates a contradiction. However, the problem states that the number per crate is a power of 2. Therefore, we must re-evaluate the initial data. Perhaps the total number of cones is not 64, or the number of crates is not 14. But the problem says: 'The total number of cones is 64, and the number of crates is 14.'
12. Given that 64 ÷ 14 = 32/7 ≈ 4.57, and 4.57 is not a power of 2, and no integer k makes 2^k = 32/7, it is mathematically impossible for 14 identical crates to hold 64 cones with each crate containing a number of cones that is a power of 2. Therefore, the problem as stated contains inconsistent data.

Reference answer: Error: The given data is inconsistent. 64 cones in 14 crates cannot result in each crate containing a number of cones that is a power of 2, as 64 ÷ 14 = 32/7, which is not a power of 2.

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