If \(\left(\frac ab\right)^{x-1}=\left(\frac ba\right)^{x-3}\), where…
2023
If \(\left(\frac ab\right)^{x-1}=\left(\frac ba\right)^{x-3}\), where \(a,b>0\) and \(a\ne b\), then the value of x is
Answer: B. 2 — ConceptFor a nonzero base, a reciprocal can be written using a negative exponent: (b/a)n = (a/b)−n. When equal powers have the same admissible base (with a/b…
- A.
1
- B.
2
- C.
\(\frac72\)
- D.
More than one of the above
- E.
None of the above
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Show answer & explanation
Correct answer: B
Concept
For a nonzero base, a reciprocal can be written using a negative exponent: (b/a)n = (a/b)−n.
When equal powers have the same admissible base (with a/b positive and not 1), their exponents are equal.
Application
Rewrite the right-hand side: (b/a)x−3 = (a/b)−(x−3) = (a/b)3−x.
The common base is now a/b, so equate the exponents: x − 1 = 3 − x.
Collect the x-terms: 2x = 4.
Divide by 2: x = 2.
Cross-check
Substituting x = 2 gives exponent 1 on the left and exponent −1 on (b/a) on the right; (b/a)−1 = a/b, so both sides are equal.
Therefore, x = 2.