Let's solve the problem step by step.

Given Expression:

$\left(\frac{\left(\frac{1}{2}\right)^3}{\left(\frac{2}{3}\right)^2}\right)^{-2}$

Step 1: Simplify the inner fraction

• Calculate $\left(\frac{1}{2}\right)^3$: $\left(\frac{1}{2}\right)^3 = \frac{1^3}{2^3} = \frac{1}{8}$

• Calculate $\left(\frac{2}{3}\right)^2$: $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$

Now, substitute these back into the fraction: $\frac{\frac{1}{8}}{\frac{4}{9}} = \frac{1}{8} \times \frac{9}{4} = \frac{9}{32}$

Step 2: Apply the outer exponent

Now raise $\frac{9}{32}$ to the power of $-2$: $\left(\frac{9}{32}\right)^{-2} = \left(\frac{32}{9}\right)^2 = \frac{32^2}{9^2} = \frac{1024}{81}$

Final Expression:

The expression simplifies to $\frac{1024}{81}$.

None of the provided options directly matches this result. However, this seems like a tricky question that may involve additional simplification or checking against possible missteps. Let's go back and check if the base was correctly managed for a multiple-choice format.

Would you like to see the detailed breakdown of how to reconcile this with the answer choices or if there's any possible approximation?