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The uploaded image seems to contain a mathematical expression involving exponents and fractions. I'll transcribe it and solve the expression step by step.

The given expression is:

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

Let's simplify this expression step by step.

### Simplifying the first term:

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

Simplify the powers of \(x\) and \(y\):

\[
= \frac{4}{3} \times x^{-3 - 2} \times y^{1 - 5} = \frac{4}{3} x^{-5} y^{-4}
\]

Now square this:

\[
\left( \frac{4}{3} x^{-5} y^{-4} \right)^2 = \frac{16}{9} x^{-10} y^{-8}
\]

### Simplifying the second term:

\[
\frac{2x^{\frac{2}{3}}y}{6x^2y^{\frac{1}{3}}}
\]

Simplify the powers of \(x\) and \(y\):

\[
= \frac{2}{6} \times x^{\frac{2}{3} - 2} \times y^{1 - \frac{1}{3}} = \frac{1}{3} x^{-\frac{4}{3}} y^{\frac{2}{3}}
\]

Now raise this to the power of \(-2\):

\[
\left( \frac{1}{3} x^{-\frac{4}{3}} y^{\frac{2}{3}} \right)^{-2} = 3^2 x^{\frac{8}{3}} y^{-\frac{4}{3}} = 9 x^{\frac{8}{3}} y^{-\frac{4}{3}}
\]

### Combining the results:

Now multiply the two simplified expressions:

\[
\frac{16}{9} x^{-10} y^{-8} \times 9 x^{\frac{8}{3}} y^{-\frac{4}{3}} 
\]

First, simplify the coefficients:

\[
\frac{16}{9} \times 9 = 16
\]

Now simplify the powers of \(x\) and \(y\):

\[
x^{-10 + \frac{8}{3}} = x^{-\frac{22}{3}}, \quad y^{-8 - \frac{4}{3}} = y^{-\frac{28}{3}}
\]

Thus, the final simplified expression is:

\[
16 x^{-\frac{22}{3}} y^{-\frac{28}{3}}
\]

Let me know if you'd like more details or further explanation! Here are some related questions:

1. What are the properties of exponents used in this simplification?
2. How do you handle negative exponents in an expression?
3. Can you explain how fractional exponents work?
4. How does the distributive property apply to powers and fractions?
5. How would this expression change if the exponents had different values?

**Tip**: When multiplying terms with exponents, always remember to add the exponents when the bases are the same.

Solve the following expression: [(4x^(-3) y) / (3x^2 y^5)]^2 * [(2x^(2/3) y) / (6x^2 y^(1/3))]^(-2)

Learn how to simplify a complex expression involving exponents, fractions, and negative exponents. This step-by-step solution covers algebraic manipulation and the laws of exponents. Suitable for students in Grades 10-12.

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