Let's work through both of the problems visible in the image.

Problem 1:

Simplify the expression using the rules of exponents:

$\frac{(-2x^2 y)^3}{9x^2 y^2}$

1. Apply the exponent: Raise the entire numerator to the power of 3: $(-2x^2 y)^3 = (-2)^3 (x^2)^3 y^3 = -8x^6 y^3$

   So the expression becomes: $\frac{-8x^6 y^3}{9x^2 y^2}$

2. Simplify the exponents:

   • For the $x$ terms: $x^6 / x^2 = x^{6-2} = x^4$
   • For the $y$ terms: $y^3 / y^2 = y^{3-2} = y^1 = y$

   The simplified expression is: $\frac{-8x^4 y}{9}$

Thus, the answer to Problem 1 is: $\boxed{\frac{-8x^4 y}{9}}$

Problem 2:

Evaluate the expression using the rules of exponents:

$2^{-4}$

1. Use the negative exponent rule: A negative exponent means taking the reciprocal of the base and making the exponent positive: $2^{-4} = \frac{1}{2^4}$

2. Calculate the power: $2^4 = 16$

Thus, the expression simplifies to: $\boxed{\frac{1}{16}}$

Would you like more details on these steps or have any specific questions?

Here are 5 related questions to help deepen your understanding:

1. How do you simplify expressions involving negative exponents in both the numerator and denominator?
2. What is the general rule for dividing terms with the same base but different exponents?
3. How can you apply exponent rules when there are multiple variables?
4. How does exponentiation affect negative numbers like $(-2)$?
5. What happens when you raise a fraction to a negative exponent?

Tip: Always simplify exponents step by step by applying the power and division rules before reducing coefficients.