Integer exponent laws

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Formula

Explanation

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Worked solution

1. The total number of training cones is 68, and they are distributed equally among 91 identical crates. The number of cones per crate is given by 68 ÷ 91.
2. This division can be written as a fraction: 68 / 91.
3. The problem states that the number of cones per crate must be expressed as a power of 91, using exponent laws. Therefore, we are to express 68 / 91 as 91 raised to some integer exponent.
4. We note that 91 = 7 × 13, and 68 = 4 × 17. Neither 68 nor 91 share common factors with 91, so 68/91 is already in simplest form and cannot be simplified to a power of 91 directly.
5. We are to use the exponent law: a^m / a^n = a^(m-n). We are given that the total cones = 68, crates = 91, and cones per crate = 68 / 91.
6. We now express 68 as 68 × 91^0, since any number to the power 0 is 1. So 68 = 68 × 91^0.
7. We write the cones per crate as (68 × 91^0) / 91^1.
8. We apply the exponent law: a^m / a^n = a^(m-n). Here, 91^0 / 91^1 = 91^(0-1) = 91^(-1).
9. So, cones per crate = 68 × 91^(-1).
10. We now check if 68 × 91^(-1) can be simplified to a single power of 91. Since 68 is not a power of 91, and 91^(-1) is 1/91, the expression 68/91 cannot be written as a single power of 91 unless 68 is a power of 91, which it is not.
11. However, the problem states to express the result as a single exact value using integer exponents, and to simplify using a^m / a^n = a^(m-n). The only valid form using exponent laws and the given data is 68 × 91^(-1).
12. Therefore, the number of cones per crate is 68 × 91^(-1), which simplifies to 68 / 91. This is the exact value, and since 68 and 91 are coprime, it cannot be reduced further. The final answer is 68 × 91^(-1).

Reference answer: 68 × 91^(-1)

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