Simplify; [($\frac{16}{9}$)$^{\frac{-3}{2}}$ x 16$^{\frac{-3}{4}}$]$^{\frac{1}{3}}$

Analyze the question: We need to simplify [(16/9)^(-3/2) × 16^(-3/4)]^(1/3).

Concept: Use the laws of indices:

• (a/b)^(-n) = (b/a)^n
• a^(m/n) = (ⁿ√a)^m
• (a^m)^n = a^(mn)

Step 1: First factor (16/9)^(-3/2) = (9/16)^(3/2) √(9/16) = 3/4, so (3/4)³ = 27/64

Step 2: Second factor 16^(-3/4) = 1/16^(3/4) 16^(1/4) = 2, so 16^(3/4) = 2³ = 8 So 16^(-3/4) = 1/8

Step 3: Multiply 27/64 × 1/8 = 27/512

Step 4: Raise to the power 1/3 (27/512)^(1/3) = 3/8, since ∛27 = 3 and ∛512 = 8.

Answer: 3/8, which is option C.

Tip: Where possible, write every number as a prime power first. Here, 16 = 2⁴ and 9 = 3², so the combined exponent on 3 is 2·(3/2)·(1/3) = 1 and on 2 is a simple fraction. This avoids big numbers like 512. Check: 3^(1) × 2^(-(4·(-3/2)·... )) works out to the same result, 3/8.