Evaluate $\left(\frac{81}{16}\right)^{\frac{-1}{4}}\times 2^{-1}$

Analyze the question: We need to evaluate (81/16)^(-1/4) × 2^(-1). It involves a negative fractional index.

Concept:

• A negative index means take the reciprocal: a^(-n) = 1/a^n
• A fractional index 1/4 means the fourth root: a^(1/4) = ⁴√a

Solution:

Step 1: Apply the negative index by taking the reciprocal. (81/16)^(-1/4) = (16/81)^(1/4)

Step 2: Take the fourth root.

• ⁴√16 = 2 (since 2⁴ = 16)
• ⁴√81 = 3 (since 3⁴ = 81)

So (16/81)^(1/4) = 2/3

Step 3: Multiply by 2^(-1) = 1/2. (2/3) × (1/2) = 2/6 = 1/3

Answer: 1/3, which is option A.

Tip: When you see a negative power on a fraction, flip the fraction first and drop the negative sign. Then deal with the root. Memorise common fourth powers: 2⁴=16, 3⁴=81, 4⁴=256, 5⁴=625.