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The given expression is:

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

Let's simplify it step by step.

1. First, simplify the fraction:
   \[
   \frac{y^3}{x^{-4}} = y^3 \cdot x^4
   \]
   (since dividing by \(x^{-4}\) is the same as multiplying by \(x^4\)).

2. Now, the expression becomes:
   \[
   \left( y^3 \cdot x^4 \right)^{-2}
   \]

3. Apply the power of a product rule, which states \((ab)^n = a^n \cdot b^n\):
   \[
   y^{3 \cdot (-2)} \cdot x^{4 \cdot (-2)} = y^{-6} \cdot x^{-8}
   \]

4. Finally, express the result with positive exponents:
   \[
   \frac{1}{y^6 \cdot x^8}
   \]

So, the simplified form of the expression is:

\[
\frac{1}{y^6 \cdot x^8}
\]

Let me know if you'd like further details or explanations!

Here are five related questions:
1. How do you simplify expressions with negative exponents?
2. What is the power of a product rule in exponents?
3. How do you handle division with exponents?
4. Can you explain the difference between a positive and a negative exponent?
5. How would this simplification change if the original exponent were positive instead of negative?

**Tip:** Always remember that a negative exponent signifies taking the reciprocal of the base raised to the corresponding positive exponent.

Simplify the expression (y^3 / x^(-4))^(-2)

Learn how to simplify the expression (y^3 / x^(-4))^(-2) using the power of a product rule and negative exponents. This solution covers the step-by-step simplification, involving core algebraic rules and exponent laws. Suitable for students in Grades 9-11.

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