Evaluate a power-of-a-power ratio: Simplify the expression [(12)^(−2)]^2 ÷ [(12)^2]^(−2) and give the exact value.

Introduction / Context:This tests exponent laws, specifically (a^m)^n = a^(m n), and how negative exponents invert the base. Carefully apply the rules to both numerator and denominator before simplifying the ratio of equal powers.

Given Data / Assumptions:

• Expression: [(12)^(−2)]^2 ÷ [(12)^2]^(−2).
• Use standard laws of indices.

Concept / Approach:Compute the power of a power for both parts, reduce each to a single power of 12, then divide. Identical powers in numerator and denominator will cancel to 1.

Step-by-Step Solution:

Verification / Alternative check:Assign 12^(−4) = 1/12^4 for both; clearly their quotient is 1.

Why Other Options Are Wrong:

• 12, 4.8, 12/144, 1/12: Result from arithmetic slips or misapplying negative exponents.

Common Pitfalls:Forgetting that raising to a negative exponent inverts, and that dividing equal powers subtracts exponents.

Final Answer:1