Evaluate a power-of-a-power ratio: Simplify the expression [(12)^(−2)]^2 ÷ [(12)^2]^(−2) and give the exact value.
Aptitude
Surds and Indices
Difficulty: Easy
Choose an option
-
A12
-
B4.8
-
C12/144
-
D1
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E1/12
Answer
Correct Answer: 1
Explanation
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:
Numerator: [(12)^(−2)]^2 = 12^(−4).Denominator: [(12)^2]^(−2) = 12^(2 * −2) = 12^(−4).Ratio = 12^(−4) / 12^(−4) = 12^(−4 − (−4)) = 12^0 = 1.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