Solve the parameter from a symmetric exponential identity: If (p/q)^(n−1) = (q/p)^(n−3), determine the value of n.
Aptitude
Surds and Indices
Difficulty: Easy
Choose an option
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A1/2
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B7/2
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C1
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D2
Answer
Correct Answer: 2
Explanation
Introduction / Context:This problem hinges on properties of exponents and reciprocals. Recognizing that (q/p) = (p/q)^(−1) converts the equation to a simple equality of exponents, allowing quick solution for n.Given Data / Assumptions:
- (p/q)^(n−1) = (q/p)^(n−3).
- Assume p and q are nonzero and p/q > 0 so real powers are defined.
Concept / Approach:Rewrite (q/p)^(n−3) as (p/q)^(−(n−3)). Since the bases match and are positive and not 1, we equate exponents directly.Step-by-Step Solution:
(p/q)^(n−1) = (p/q)^(−(n−3)).Therefore, n − 1 = −(n − 3).n − 1 = −n + 3 ⇒ 2n = 4 ⇒ n = 2.Verification / Alternative check:Substitute n = 2: LHS = (p/q)^(1) = p/q; RHS = (q/p)^(−1) = p/q. Both sides match.
Why Other Options Are Wrong:
- 1/2, 7/2, 1: These do not satisfy the exponent equation and fail upon substitution.
Common Pitfalls:Forgetting that (q/p) = (p/q)^(−1) or mishandling the minus sign when moving exponents across the equality.
Final Answer:
2