If 27^x + 27^{x − 1/3} = 972, find the value of x.

Introduction / Context:

This exponential equation question tests your ability to factor out a common power and to recognise numbers written as powers of a smaller base. The base 27 suggests that we should use base 3, since 27 is 3^3, and that the equation can be simplified significantly by a substitution. This is a standard pattern in exponent problems in aptitude tests.

Given Data / Assumptions:

• The equation is 27^x + 27^{x − 1/3} = 972.
• All exponents and numbers are real and positive.
• You must determine the real value of x that satisfies this equation.

Concept / Approach:

The main strategy is to factor out the smaller exponential term. Specifically, you can factor out 27^{x − 1/3} from both terms on the left. This gives a simple linear expression in terms of that common factor. Then, by writing 27 and 972 as powers of 3, you can solve for x using basic rules of exponents. Recognising 972 as 27 times 36 and 243 as 3^5 is helpful.

Step-by-Step Solution:

Verification / Alternative check:

Substitute x = 2 back into the original equation. Compute 27^2 = 729 and 27^{2 − 1/3} = 27^{5/3} = (27^{1/3})^5 = 3^5 = 243. Then 729 + 243 = 972, which matches the right side. This confirms that x = 2 is the correct solution.

Why Other Options Are Wrong:

The values 3, 4, 5, and 1 do not satisfy the original equation. If x were 3, 27^3 is already much larger than 972. If x were 1, the left side becomes 27 + 27^{2/3}, which is less than 972. Only x = 2 makes the sum exactly 972, consistent with both the algebraic solution and the numerical check.

Common Pitfalls:

Some students try to take logarithms immediately, which is unnecessary and can complicate the algebra. Others forget to factor out the common term correctly or miscalculate 972 / 4. Recognising patterns and working systematically with powers of 3 is the fastest and most reliable approach.

Final Answer:

The value of x that satisfies the equation is 2.