Use exponent rules to rewrite both sides with the same base.\nIf (2^3)^2 = 4^x, what is the value of 3x?

Introduction / Context:
This question tests basic exponent laws: power of a power, and rewriting numbers with the same base. The goal is to compare exponents once both sides are written as powers of 2. After finding x, we simply compute 3x.

Given Data / Assumptions:

• (2^3)^2 = 4^x
• We need the value of 3x.
• Use rules: (a^m)^n = a^(m*n) and 4 = 2^2.

Concept / Approach:
Rewrite both sides with base 2. Then equate exponents because if 2^p = 2^q, then p = q. Finally multiply x by 3 to match the question requirement.

Step-by-Step Solution:
Left side: (2^3)^2 = 2^(3*2) = 2^6 Right side: 4^x = (2^2)^x = 2^(2x) So the equation becomes: 2^6 = 2^(2x) Equate exponents: 6 = 2x Solve: x = 3 Compute 3x: 3x = 3*3 = 9

Verification / Alternative check:
Check numerically: (2^3)^2 = 8^2 = 64. If x = 3, then 4^x = 4^3 = 64. Both sides match, so x is correct and 3x = 9 is confirmed.

Why Other Options Are Wrong:
6 is the exponent on 2 in 2^6, but the question asks for 3x, not 2x or 6.
12 comes from incorrectly doing 3x = 2x + something, or assuming x = 4.
27 comes from mistakenly cubing instead of multiplying by 3.
3 comes from incorrectly taking x itself as the final answer without multiplying by 3.

Common Pitfalls:
Forgetting that 4 = 2^2, or misusing (a^m)^n as a^(m+n) instead of a^(m*n). Also, some learners stop after finding x and forget to compute 3x.

Final Answer:
3x = 9