https://r2.erweima.ai/mathbot/image/1725813580842rmrioxat.jpg

The expression in the image can be written as follows:

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

Let's simplify each term step by step.

### Simplifying the first part:
\[
\frac{4x^{-3}y^2}{3x^2y^5}
\]

For the constants:
\[
\frac{4}{3}
\]

For the powers of \(x\):
\[
x^{-3} \div x^2 = x^{-3 - 2} = x^{-5}
\]

For the powers of \(y\):
\[
y^2 \div y^5 = y^{2 - 5} = y^{-3}
\]

So, the first part becomes:
\[
\frac{4}{3} x^{-5} y^{-3}
\]

Raising this to the power of \(-2\):
\[
\left(\frac{4}{3} x^{-5} y^{-3}\right)^{-2} = \frac{3^2}{4^2} x^{10} y^6 = \frac{9}{16} x^{10} y^6
\]

### Simplifying the second part:
\[
\frac{2x^{\frac{2}{3}}y}{6x^2y^{\frac{1}{3}}}
\]

For the constants:
\[
\frac{2}{6} = \frac{1}{3}
\]

For the powers of \(x\):
\[
x^{\frac{2}{3}} \div x^2 = x^{\frac{2}{3} - 2} = x^{-\frac{4}{3}}
\]

For the powers of \(y\):
\[
y \div y^{\frac{1}{3}} = y^{1 - \frac{1}{3}} = y^{\frac{2}{3}}
\]

So, the second part becomes:
\[
\frac{1}{3} x^{-\frac{4}{3}} y^{\frac{2}{3}}
\]

Raising this to the power of \(2\):
\[
\left(\frac{1}{3} x^{-\frac{4}{3}} y^{\frac{2}{3}}\right)^2 = \frac{1^2}{3^2} x^{-\frac{8}{3}} y^{\frac{4}{3}} = \frac{1}{9} x^{-\frac{8}{3}} y^{\frac{4}{3}}
\]

### Final Expression:

Now, multiplying both results together:
\[
\frac{9}{16} x^{10} y^6 \times \frac{1}{9} x^{-\frac{8}{3}} y^{\frac{4}{3}} = \frac{9 \times 1}{16 \times 9} x^{10 - \frac{8}{3}} y^{6 + \frac{4}{3}} = \frac{1}{16} x^{\frac{30}{3} - \frac{8}{3}} y^{\frac{18}{3} + \frac{4}{3}} = \frac{1}{16} x^{\frac{22}{3}} y^{\frac{22}{3}}
\]

Thus, the simplified expression is:
\[
\frac{1}{16} x^{\frac{22}{3}} y^{\frac{22}{3}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you handle negative exponents in algebraic expressions?
2. What is the general process for simplifying fractions with exponents?
3. How does raising a fraction to a power affect both the numerator and denominator?
4. Why do we add or subtract exponents when dividing or multiplying like bases?
5. How do fractional exponents relate to radicals (roots)?

**Tip:** When simplifying expressions with exponents, always handle constants and variables separately to avoid mistakes.

Simplify the expression given in the image, which involves exponents and fractions.

This problem involves simplifying a complex algebraic expression with exponents and fractions. It covers key concepts such as the laws of exponents and the process of raising fractions to a power. Suitable for high school students in Grades 10-12.

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