The given expression appears to be a complex algebraic expression involving powers, roots, and products of variables $x$ and $y$. Let's break it down step-by-step.

The expression is:

$\left\{ x^{\frac{2}{3}} \left( x^{-\frac{1}{3}} y^{\frac{1}{2}} \left( x^2 y^{-2} \right)^{-\frac{2}{3}} \right)^{\frac{1}{2}} \right\}^6$

Steps to simplify:

1. Simplify the innermost term $(x^2 y^{-2})^{-\frac{2}{3}}$: $(x^2 y^{-2})^{-\frac{2}{3}} = x^{2 \cdot -\frac{2}{3}} y^{-2 \cdot -\frac{2}{3}} = x^{-\frac{4}{3}} y^{\frac{4}{3}}$

2. Substitute this result into the expression: $x^{-\frac{1}{3}} y^{\frac{1}{2}} \left( x^{-\frac{4}{3}} y^{\frac{4}{3}} \right)$ Simplify the product of powers: $= x^{-\frac{1}{3} - \frac{4}{3}} y^{\frac{1}{2} + \frac{4}{3}} = x^{-\frac{5}{3}} y^{\frac{11}{6}}$

3. Now take the square root $\left( x^{-\frac{5}{3}} y^{\frac{11}{6}} \right)^{\frac{1}{2}}$: $= x^{-\frac{5}{6}} y^{\frac{11}{12}}$

4. Multiply this result by $x^{\frac{2}{3}}$: $x^{\frac{2}{3}} \cdot x^{-\frac{5}{6}} y^{\frac{11}{12}} = x^{\frac{4}{6} - \frac{5}{6}} y^{\frac{11}{12}} = x^{-\frac{1}{6}} y^{\frac{11}{12}}$

5. Finally, raise the entire expression to the power of 6: $\left( x^{-\frac{1}{6}} y^{\frac{11}{12}} \right)^6 = x^{-1} y^{\frac{11}{2}}$

Final Answer:

$x^{-1} y^{\frac{11}{2}}$

Would you like a more detailed explanation on any of these steps?

Here are 5 related questions for further exploration:

1. How would the result change if one of the exponents was different?
2. Can this expression be simplified further if additional terms are introduced?
3. How would this result change if the variables $x$ and $y$ were specific numbers?
4. What are the rules governing exponent operations in expressions like these?
5. How can we use logarithms to further analyze complex exponential expressions?

Tip: Always check if all variables have positive exponents at the end to simplify interpretation.