Negative and fractional exponents: Simplify (x^(2/3))^(−3/4) and express it with positive exponents if possible.

Introduction / Context:This checks the power-of-a-power rule and handling negative exponents. We convert the nested exponent to a single exponent on x, then rewrite with a positive exponent as a reciprocal if needed.

Given Data / Assumptions:

• Expression: (x^(2/3))^(−3/4)
• x > 0 for real radical interpretation.

Concept / Approach:Use (a^m)^n = a^(mn). Multiply exponents 2/3 and −3/4 to get a single exponent. A negative exponent −k means 1/x^k. Convert x^(−1/2) to 1/√x.

Step-by-Step Solution:(x^(2/3))^(−3/4) = x^((2/3)*(−3/4)) = x^(−6/12) = x^(−1/2)x^(−1/2) = 1 / x^(1/2) = 1/√x

Verification / Alternative check:Test x = 16: LHS = (16^(2/3))^(−3/4). 16^(2/3) = (2^4)^(2/3) = 2^(8/3). Raising to −3/4 yields 2^(−2) = 1/4. RHS 1/√16 = 1/4—matches.

Why Other Options Are Wrong:1/x and 1/x^2 correspond to exponents −1 and −2; 1/x^(−2) simplifies to x^2; √x is the reciprocal of the answer.

Common Pitfalls:Adding instead of multiplying exponents inside a power-of-a-power; mishandling negative signs.

Final Answer:1/√x