Final answer:
The question seems to contain an error as the simplified expression 7^-43 / 13 does not match any of the provided options a) 7^50, b) 7^-50, c) 7^48, or d) 7^-48, which may suggest a typo or mistake in the question itself.
Step-by-step explanation:
The student's question asks to calculate the quotient of (7^-4/7 ÷ (1/7)^13 ÷ (1/7)^5 ÷ 7^57 · 13). To solve this, we can use the properties of exponents and division. First, we simplify the expression by noticing that in the denominator, we have three terms with a base of 7, which can be combined, and also that dividing by a number is the same as multiplying by its reciprocal.
The expression simplifies to —
7^-4 × 7^-13 × 7^-5 × (1/7)^57 × (1/13). Next, combine the exponents of the same base by addition or subtraction, which in this case, is subtraction since we're dividing. We get — 7^-4 - (-13) - (-5) - 57, which simplifies to 7^-4 + 13 + 5 - 57 = 7^-43.
Finally, simplifying further, we multiply by 1/13. Since 13 is not a power of 7, it remains outside, giving us the simplified expression 7^-43 / 13. This is none of the provided options, so there seems to be an error in the original question or the provided choices.