Evaluate the power tower with surds: Compute the exact value of [(√2)^(√2)]^(√2). Choose the best classification of the result.

Introduction / Context:
Exponent laws remain valid for positive real bases and real exponents: (a^b)^c = a^(b*c). This problem applies that idea to a surd base √2 and two surd exponents √2. The goal is to simplify the nested exponents and then classify the final value in the standard number hierarchy.

Given Data / Assumptions:

• Expression: [(√2)^(√2)]^(√2).
• Base a = √2 > 0, so exponent laws for real exponents apply.
• All operations are real-valued.

Concept / Approach:
Use the law (a^b)^c = a^(b*c). Multiplying the two √2 exponents gives √2 * √2 = 2. Thus the entire tower collapses to (√2)^2, which is an exact integer. We then identify its set membership: integer, natural, rational, real.

Step-by-Step Solution:

Verification / Alternative check:
Compute numerically: √2 ≈ 1.4142. Then (√2)^(√2) ≈ 1.4142^1.4142 ≈ 1.632..., and raising that to √2 returns exactly 2 by the algebraic identity used.

Why Other Options Are Wrong:
The value 2 is an integer and a natural number (in the usual positive-integer convention). Therefore it is not “non-integer rational” nor “irrational.”

Common Pitfalls:
Believing that a surd base automatically produces an irrational result after exponentiation. Here, the structure of exponents forces an exact integer after simplification.

Final Answer:
a natural number