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The image shows an equation involving exponents, and the task is to select the equivalent expression from the options provided. The expression in the image is:

\[
(e^x \cdot y^3)^{-1} \cdot x^{-2}
\]

The options given are:
1. \( y^{-3} \cdot x^2 \)
2. \( (xy)^{-3} \)
3. \( 2y^3 \cdot x^{-1} \)

### Step-by-Step Solution:

1. **Start by simplifying the expression**:
   \[
   (e^x \cdot y^3)^{-1} \cdot x^{-2}
   \]
   
2. **Apply the power of -1 to both factors** inside the parentheses:
   \[
   (e^x)^{-1} \cdot (y^3)^{-1} \cdot x^{-2}
   \]
   
   Using exponent rules:
   \[
   (e^x)^{-1} = e^{-x} \quad \text{and} \quad (y^3)^{-1} = y^{-3}
   \]
   
   So the expression becomes:
   \[
   e^{-x} \cdot y^{-3} \cdot x^{-2}
   \]
   
3. **Combine like terms** if possible, though here they are distinct. Thus, the simplified form of the expression is:
   \[
   e^{-x} \cdot y^{-3} \cdot x^{-2}
   \]

4. **Match this with the given options**:
   - Option 1: \( y^{-3} \cdot x^2 \) (this is close but not quite right since it doesn't include the \( e^{-x} \))
   - Option 2: \( (xy)^{-3} \) (not equivalent due to mismatched exponents)
   - Option 3: \( 2y^3 \cdot x^{-1} \) (not a match either)

None of the options directly matches the full expression \( e^{-x} \cdot y^{-3} \cdot x^{-2} \). If we ignore the \( e^{-x} \) term and look at only the variables \( y \) and \( x \), Option 1 is the closest match after simplifying the powers. 

### Final Answer:
The closest equivalent expression is:
\[
\boxed{1. \, y^{-3} \cdot x^2}
\]

Would you like further details on the steps or have any other questions?

### Related Questions:
1. What are the exponent rules applied in simplifying powers?
2. How do we handle negative exponents when simplifying expressions?
3. How can we further expand this expression if we include \( e^x \)?
4. What happens if we distribute exponents to multiple terms inside parentheses?
5. What are common mistakes to avoid when simplifying expressions with exponents?

### Tip:
Always apply exponent rules carefully, especially when dealing with negative exponents and multiple factors within parentheses.

Select the equivalent expression: (e^x * y^3)^-1 * x^-2. Choose 1 answer from the given options.

Explore the process of simplifying expressions involving exponents with a detailed breakdown of the expression (e^x * y^3)^-1 * x^-2. Learn the key exponent rules and simplification techniques suitable for Grades 9-10 students.

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