What is the numerical value of 2 raised to the power (3^2), that is 2^(3^2)?

Introduction / Context:
This question checks understanding of exponents and the correct interpretation of powers written in a compact format. The expression is 2^(3^2), meaning 2 raised to the power 3 squared, not (2^3)^2 unless explicitly bracketed that way. Recognising the order of evaluation in exponent towers is crucial in such problems.

Given Data / Assumptions:

- The base is 2.

- The exponent is 3^2.

- 3^2 means 3 raised to the power 2.

- We must compute 2^(3^2).

Concept / Approach:
First evaluate the inner exponent 3^2. Then use that result as the power of 2. Exponentiation is performed from the top down in an exponent tower: 2^(3^2) is 2^9, not (2^3)^2. Once we have 2^9, we can compute it step by step using repeated doubling.

Step-by-Step Solution:
Step 1: Compute 3^2. 3^2 = 3 * 3 = 9. Step 2: Rewrite the expression using this value. 2^(3^2) = 2^9. Step 3: Compute 2^9. 2^1 = 2. 2^2 = 4. 2^3 = 8. 2^4 = 16. 2^5 = 32. 2^6 = 64. 2^7 = 128. 2^8 = 256. 2^9 = 512.

Verification / Alternative check:
You can group powers to check: 2^9 = (2^3)^3 = 8^3 = 8 * 8 * 8 = 64 * 8 = 512. This matches our earlier calculation, so the value 512 is consistent and confirmed.

Why Other Options Are Wrong:
- 64: This is 2^6, which would come from misreading the expression.

- 128: This equals 2^7, another incorrect power.

- 256: This equals 2^8, which is still below 2^9.

None of these match 2^9.

Common Pitfalls:
A common mistake is to misinterpret the notation and compute (2^3)^2 = 2^6 = 64 instead of 2^(3^2). Another error is to stop early when computing powers of 2 by hand and miscount the steps. Careful reading and systematic computation prevent these mistakes.

Final Answer:
The value of 2^(3^2) is 512.