Difficulty: Easy
Correct Answer: 17
Explanation:
Introduction / Context:This exercise tests exponent rules and base conversion. The idea is to combine products and quotients of powers of 4, then rewrite the final result as a power of 64. Recognizing that 64 = 4^3 is the shortcut to matching bases.
Given Data / Assumptions:
Concept / Approach:Apply exponent rules: (a^m)^n = a^(mn) and a^p × a^q = a^(p+q). For division, a^p ÷ a^q = a^(p−q). After simplifying to a single power of 4, convert to a power of 64 using base equivalence.
Step-by-Step Solution:
(4^6)^5 = 4^(6*5) = 4^30.(4^3)^8 = 4^(3*8) = 4^24.Product: 4^30 × 4^24 = 4^(30+24) = 4^54.Divide by 4^3 → 4^(54 − 3) = 4^51.Since 64 = 4^3, write 4^51 = (4^3)^(17) = 64^17.Verification / Alternative check:Reverse substitution: 64^17 = (4^3)^17 = 4^(51), exactly matching the simplified exponent on 4. Hence ? = 17.
Why Other Options Are Wrong:10, 11, 15, and 16 do not satisfy 3×? = 51. Only 17 makes 64^? equal to 4^51.
Common Pitfalls:Adding instead of multiplying exponents in (a^m)^n, or forgetting to subtract exponents when dividing like bases.
Final Answer:17
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