Exponent equivalence: Evaluate (4×4×4×4×4×4)^5 × (4×4×4)^8 ÷ (4)^3 and express the result as (64)^?. Determine the integer exponent ?.

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:

• (4×4×4×4×4×4) = 4^6.
• (4×4×4) = 4^3.
• Expression: (4^6)^5 × (4^3)^8 ÷ 4^3.
• Target form: (64)^? with 64 = 4^3.

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:

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