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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?

Simplifique y exprese el resultado en forma de fracción: \left(\frac{\left(\frac{1}{2}\right)^3}{\left(\frac{2}{3}\right)^2}\right)^{-2}

Learn how to simplify complex fraction expressions involving exponents. This problem applies key concepts such as exponent rules and fraction operations to solve the expression \left(\frac{\left(\frac{1}{2}\right)^3}{\left(\frac{2}{3}\right)^2}\right)^{-2}. Suitable for students in Grades 8-10.

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