If (√2)^n = 64, determine the value of n. (Interpretation: the expression denotes (√2) raised to the power n equals 64.)

Introduction / Context: Exponent rules are central in this problem. Recognizing powers of 2 and rewriting square roots in exponential form lead to a quick solution.

Given Data / Assumptions:

• (√2)^n = 64.
• 64 = 2^6.
• √2 = 2^(1/2).

Concept / Approach: Convert √2 to exponential form, then equate exponents for equal bases. If 2^(n/2) = 2^6, then n/2 = 6 and n follows immediately.

Step-by-Step Solution: √2 = 2^(1/2). (√2)^n = (2^(1/2))^n = 2^(n/2). Given 2^(n/2) = 64 = 2^6. Therefore n/2 = 6 ⇒ n = 12.

Verification / Alternative check: Substitute back: (√2)^12 = (2^(1/2))^12 = 2^6 = 64. Correct.

Why Other Options Are Wrong: 2, 4, 6 do not satisfy 2^(n/2) = 2^6.

Common Pitfalls: Misreading √2^n as √(2^n) rather than (√2)^n changes the equation. The wording here clarifies it should be interpreted as (√2)^n.

Final Answer: 12