Integer exponent laws

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Formula

Explanation

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Worked solution

1. The total number of training balls is 55, and they are to be distributed equally among 13 identical storage units.
2. The number of balls per unit is calculated as 55 ÷ 13.
3. This gives 55/13 balls per unit, which is approximately 4.23, but we must express it as a power of 13 using exponent laws.
4. We are told the number of balls per unit must be expressed as a power of 13. So we write 55/13 as 13^m for some integer m.
5. We know 55 = 5 × 11 and 13 is prime, so 55/13 cannot be simplified to a power of 13 unless 55 is a multiple of a power of 13.
6. 13^1 = 13, 13^2 = 169, which is greater than 55. So 13^1 is too small, 13^2 is too big. No positive integer power of 13 equals 55/13.
7. But 55/13 = 55 × 13^(-1). We now express 55 as a power of 13, but 55 is not a power of 13.
8. We are to use exponent laws: a^m / a^n = a^(m-n). We write 55/13 as 55 × 13^(-1).
9. Since 55 is not a power of 13, we cannot write 55 as 13^k for any integer k. Therefore, 55/13 cannot be written as a single power of 13 unless 55 itself is a power of 13, which it is not.
10. However, the problem states that the number of balls per unit is expressed as a power of 13. This implies that 55 must be written as 13^a × 13^b = 13^(a+b), but 55 is not a power of 13.
11. We reconsider: 55 ÷ 13 = 55/13. We write this as (55) × 13^(-1). But 55 is not a power of 13, so 55/13 cannot be simplified to a single power of 13.
12. The only way 55/13 can be expressed as a power of 13 is if 55 = 13^k for some k, but 13^1 = 13, 13^2 = 169, so 55 is not a power of 13. Therefore, it is impossible to express 55/13 as a single power of 13 using integer exponents.

Reference answer: impossible to express as a single power of 13

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