The mathematical expression in the image can be interpreted as follows:

$\left[ \frac{\left( 3x^{-2}y^3 \right)^4 \left( x^5 y^{-6} \right)^{-6}}{9x^{-4}y^2} \right]^{-2}$

Step-by-step breakdown:

1. Inside the numerator:

   • The first term is $\left( 3x^{-2}y^3 \right)^4$, which we expand by raising both the coefficients and variables to the 4th power: $\left( 3x^{-2}y^3 \right)^4 = 3^4 \cdot x^{-8} \cdot y^{12} = 81x^{-8}y^{12}$

   • The second term is $\left( x^5 y^{-6} \right)^{-6}$, which involves raising each variable to the power of $-6$: $\left( x^5 y^{-6} \right)^{-6} = x^{5(-6)} \cdot y^{-6(-6)} = x^{-30} \cdot y^{36}$

   So, multiplying these together: $81x^{-8}y^{12} \cdot x^{-30}y^{36} = 81x^{-38}y^{48}$

2. In the denominator: The expression is $9x^{-4}y^2$.

3. Now, simplify the fraction:

   $\frac{81x^{-38}y^{48}}{9x^{-4}y^2} = \frac{81}{9} \cdot x^{-38 + 4} \cdot y^{48 - 2} = 9x^{-34}y^{46}$

4. Apply the outer exponent $-2$:

   $\left( 9x^{-34}y^{46} \right)^{-2} = 9^{-2} \cdot x^{-34(-2)} \cdot y^{46(-2)} = \frac{1}{81} \cdot x^{68} \cdot y^{-92}$

Thus, the simplified expression is:

$\boxed{\frac{x^{68}}{81y^{92}}}$

Would you like more details or have any questions about the steps?

5 Related Questions:

1. How do you handle negative exponents in a more complex expression?
2. What are the rules for distributing exponents across products or quotients?
3. How do fractional exponents relate to roots (e.g., square roots)?
4. How can you simplify an expression that includes multiple variables raised to various powers?
5. What is the effect of raising an entire fraction to a negative power?

Tip:

Always remember to apply exponent rules carefully when dealing with products, quotients, and nested exponents to avoid mistakes in simplification.