Solve an index equation with a radical exponent: If √(2^n) = 64, find the value of n.

Introduction / Context:Combining radicals with exponents requires converting everything to exponent form. Noting that √(2^n) = (2^n)^(1/2) = 2^(n/2) allows direct comparison to 64, a known power of 2.Given Data / Assumptions:

• √(2^n) = 64.
• 64 = 2^6.

Concept / Approach:Rewrite the radical as an exponent, equate powers of 2, and solve for n linearly. This is a standard exponent-matching technique used throughout indices problems.Step-by-Step Solution:

Verification / Alternative check:Check directly: 2^12 = 4096; √4096 = 64. Correct.

Why Other Options Are Wrong:

• 8, 4, 16: Yield √(2^n) values different from 64 (2^4=16 → √16=4; 2^8=256 → √256=16; 2^16 too large).

Common Pitfalls:Forgetting that √(a) = a^(1/2) or misidentifying 64 as 2^5 (it is 2^6).

Final Answer: