If 3^x · 27^(x−1) = 243, what is the value of x?
Answer and explanation
Correct answer: 2
The governing concept is conversion to a common base and then comparison of exponents. Since 27 = 3^3, we have 27^(x−1) = (3^3)^(x−1) = 3^(3x−3). Therefore the left-hand side becomes 3^x · 3^(3x−3) = 3^(4x−3). Also, 243 = 3^5. The bases are equal and the base 3 is positive and different from 1, so their exponents must be equal: 4x − 3 = 5. Adding 3 gives 4x = 8, and dividing by 4 gives x = 2. Substitution confirms the result: 3^2 · 27^1 = 9 · 27 = 243. Hence option B is correct. Option A leaves the exponent too small, while C and D do not satisfy the original equation.
Frequently asked questions
What is the correct answer to this question?
2
Why is this the correct answer?
The governing concept is conversion to a common base and then comparison of exponents. Since 27 = 3^3, we have 27^(x−1) = (3^3)^(x−1) = 3^(3x−3). Therefore the left-hand side becomes 3^x · 3^(3x−3) = 3^(4x−3). Also, 243 = 3^5. The bases are equal and the base 3 is positive and different from 1, so their exponents must be equal: 4x − 3 = 5. Adding 3 gives 4x = 8, and dividing by 4 gives x = 2. Substitution confirms the result: 3^2 · 27^1 = 9 · 27 = 243. Hence option B is correct. Option A leaves the exponent too small, while C and D do not satisfy the original equation.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Operations on real numbers and the laws of exponents.
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