The expression in the image appears to be:

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

Step-by-step solution:

1. Simplifying the inner parentheses:

   Inside the parentheses, we have: $x^2 y^{-2}$

   Since there's no additional simplification here, we leave this as is.

2. Applying the exponent of $-\frac{2}{3}$ to $x^2 y^{-2}$:

   Now, raise each factor inside the parentheses to the power of $-\frac{2}{3}$: $\left( x^2 \right)^{-\frac{2}{3}} = x^{2 \times -\frac{2}{3}} = x^{-\frac{4}{3}}$ $\left( y^{-2} \right)^{-\frac{2}{3}} = y^{-2 \times -\frac{2}{3}} = y^{\frac{4}{3}}$

   So, the expression inside the parentheses becomes: $x^{-\frac{4}{3}} y^{\frac{4}{3}}$

3. Multiplying this result by $x^{-\frac{1}{3}} y^{-\frac{1}{2}}$:

   We now multiply the result of step 2 by $x^{-\frac{1}{3}} y^{-\frac{1}{2}}$: $x^{-\frac{1}{3}} \times x^{-\frac{4}{3}} = x^{-\frac{1}{3} - \frac{4}{3}} = x^{-\frac{5}{3}}$ $y^{-\frac{1}{2}} \times y^{\frac{4}{3}} = y^{-\frac{1}{2} + \frac{4}{3}}$ To simplify $-\frac{1}{2} + \frac{4}{3}$, find a common denominator: $-\frac{1}{2} = -\frac{3}{6}, \quad \frac{4}{3} = \frac{8}{6}$ So: $-\frac{1}{2} + \frac{4}{3} = \frac{8}{6} - \frac{3}{6} = \frac{5}{6}$ Thus, the product is: $y^{\frac{5}{6}}$

   Therefore, the expression becomes: $x^{-\frac{5}{3}} y^{\frac{5}{6}}$

4. Applying the exponent of $\frac{1}{2}$:

   Now, we apply the exponent of $\frac{1}{2}$ to the entire expression: $\left( x^{-\frac{5}{3}} y^{\frac{5}{6}} \right)^{\frac{1}{2}}$ This gives: $x^{-\frac{5}{3} \times \frac{1}{2}} = x^{-\frac{5}{6}}$ $y^{\frac{5}{6} \times \frac{1}{2}} = y^{\frac{5}{12}}$

   So, the result is: $x^{-\frac{5}{6}} y^{\frac{5}{12}}$

5. Multiplying by $x^{\frac{2}{3}}$:

   Finally, multiply this by $x^{\frac{2}{3}}$: $x^{\frac{2}{3}} \times x^{-\frac{5}{6}} = x^{\frac{2}{3} - \frac{5}{6}}$ Find a common denominator: $\frac{2}{3} = \frac{4}{6}, \quad \frac{2}{3} - \frac{5}{6} = \frac{4}{6} - \frac{5}{6} = -\frac{1}{6}$ Thus: $x^{-\frac{1}{6}}$ Therefore, the final expression is: $x^{-\frac{1}{6}} y^{\frac{5}{12}}$

Final Answer:

$x^{-\frac{1}{6}} y^{\frac{5}{12}}$

Would you like more details, or do you have any questions?

Related Questions:

1. How do you apply exponents to products of terms?
2. What are the rules for adding and subtracting exponents with the same base?
3. How do you handle fractional exponents?
4. What is the process for simplifying complex exponents in algebraic expressions?
5. How do negative exponents affect the base term?

Tip: Always simplify the terms inside parentheses before applying outer exponents to avoid confusion in longer algebraic expressions.