https://r2.erweima.ai/mathbot/image/1725844275946s9yg4fpg.jpg

The expression provided in the image is:

\[
\left[\frac{(4x^{-3}y^2)^2}{(3x^2y^5)}\right]^{-2} \left[\frac{(2x^{\frac{2}{3}}y)}{(6x^2y^{\frac{1}{3}})}\right]^2
\]

Let's simplify this step by step.

### Simplification Steps:

1. **Simplify each bracketed term:**

   - For the first term:
     \[
     \left[\frac{(4x^{-3}y^2)^2}{3x^2y^5}\right]^{-2}
     \]
     Simplify the numerator:
     \[
     (4x^{-3}y^2)^2 = 16x^{-6}y^4
     \]
     The expression becomes:
     \[
     \frac{16x^{-6}y^4}{3x^2y^5} = \frac{16x^{-6-2}y^{4-5}}{3} = \frac{16x^{-8}y^{-1}}{3}
     \]
     Simplifying further:
     \[
     \left[\frac{16x^{-8}y^{-1}}{3}\right]^{-2} = \left(\frac{3}{16x^{-8}y^{-1}}\right)^2 = \frac{9}{256x^{-16}y^{-2}}
     \]
     Simplify the expression:
     \[
     \frac{9x^{16}y^2}{256}
     \]

   - For the second term:
     \[
     \left[\frac{2x^{\frac{2}{3}}y}{6x^2y^{\frac{1}{3}}}\right]^2
     \]
     Simplify:
     \[
     \frac{2x^{\frac{2}{3}}y}{6x^2y^{\frac{1}{3}}} = \frac{2}{6} \cdot \frac{x^{\frac{2}{3}-2}y^{1-\frac{1}{3}}}{1} = \frac{1}{3} \cdot \frac{x^{\frac{2}{3}-\frac{6}{3}}y^{\frac{2}{3}}}{1} = \frac{x^{-\frac{4}{3}}y^{\frac{2}{3}}}{3}
     \]
     Simplify further:
     \[
     \left(\frac{x^{-\frac{4}{3}}y^{\frac{2}{3}}}{3}\right)^2 = \frac{x^{-\frac{8}{3}}y^{\frac{4}{3}}}{9}
     \]

2. **Multiply the simplified expressions:**
   \[
   \frac{9x^{16}y^2}{256} \cdot \frac{x^{-\frac{8}{3}}y^{\frac{4}{3}}}{9}
   \]
   Simplify further:
   \[
   \frac{x^{16-\frac{8}{3}}y^{2+\frac{4}{3}}}{256}
   \]
   Simplify the exponents:
   \[
   16 - \frac{8}{3} = \frac{48}{3} - \frac{8}{3} = \frac{40}{3}, \quad 2 + \frac{4}{3} = \frac{6}{3} + \frac{4}{3} = \frac{10}{3}
   \]
   So the expression becomes:
   \[
   \frac{x^{\frac{40}{3}}y^{\frac{10}{3}}}{256}
   \]

**Final Simplified Expression:**
\[
\frac{x^{\frac{40}{3}}y^{\frac{10}{3}}}{256}
\]

Would you like more details on any of the steps, or do you have any specific questions?

Here are 5 related questions you might explore:
1. How would the expression change if the powers were different?
2. How can we simplify expressions involving negative exponents?
3. What are the rules for multiplying and dividing terms with exponents?
4. How does the power of a fraction affect its numerator and denominator?
5. What is the significance of fractional exponents?

**Tip:** Always simplify exponents by combining like terms and reducing fractions to their simplest form.

Simplify the expression in the image containing exponents and variables.

This page walks through the detailed step-by-step simplification of a complex expression involving exponents and variables. Using exponent rules, negative exponents, and rational expressions, the solution explains each step clearly for students in Grades 9-12.

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