Solve an exponential equation and evaluate x^x: If 3^x − 3^(x−1) = 18, find the value of x^x.
Aptitude
Surds and Indices
Difficulty: Easy
Choose an option
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A3
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B8
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C27
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D216
Answer
Correct Answer: 27
Explanation
Introduction / Context:The equation features the same base 3 raised to related exponents. Factoring out the common term 3^(x−1) reduces it to a simple linear equation in the exponent. Then evaluate x^x using the obtained integer x.Given Data / Assumptions:
- 3^x − 3^(x−1) = 18.
- Compute x^x after solving x.
Concept / Approach:Use 3^x = 3 * 3^(x−1). Then the left side becomes (3 − 1)*3^(x−1) = 2*3^(x−1), which is easy to solve.Step-by-Step Solution:
3^x − 3^(x−1) = 3^(x−1)(3 − 1) = 2*3^(x−1) = 18.3^(x−1) = 9 = 3^2 ⇒ x − 1 = 2 ⇒ x = 3.Compute x^x = 3^3 = 27.Verification / Alternative check:Substitute x = 3 into the original: 27 − 9 = 18, correct.
Why Other Options Are Wrong:
- 3, 8, 216: These correspond to 3^1, 2^3, and 6^3—unrelated to x^x with x = 3.
Common Pitfalls:Not factoring out 3^(x−1) or misapplying exponent subtraction.
Final Answer:
27