Exponent rules and base conversion: Evaluate (4^6)^5 × (4^3)^8 ÷ 4^3 and express the result as (64)^k. Find k.

Introduction / Context:This problem checks familiarity with exponent laws and rewriting a power of 4 in terms of base 64. The goal is to simplify the left-hand side into a single power of 4 and then equate it to a power of 64 to find k.

Given Data / Assumptions:

• Expression: (4^6)^5 × (4^3)^8 ÷ 4^3
• Target form: (64)^k
• Recall: (a^m)^n = a^(m*n), a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n)

Concept / Approach:Simplify the left-hand side to 4^power. Then use 64 = 4^3 to convert 4^power into (4^3)^k = 4^(3k) and solve 3k = power.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 10/11/16: Result from arithmetic slips (e.g., adding exponents incorrectly or forgetting the final division by 4^3).

Common Pitfalls:

• Misapplying (a^m)^n as a^(m + n) instead of a^(m*n).
• Not converting 64 to 4^3 before equating exponents.

Final Answer: