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The expression provided is:

\[
\left( \frac{-3x^2 y^5}{2y^2 x^7} \right)^3
\]

To simplify this expression, we’ll proceed in steps:

### Step 1: Simplify Inside the Parentheses
Inside the parentheses, we have:

\[
\frac{-3x^2 y^5}{2y^2 x^7}
\]

Let's handle the coefficients and variables separately:

1. **Coefficient**: The coefficient is \(\frac{-3}{2}\).
2. **Variable \(x\)**: The powers of \(x\) in the numerator and denominator are \(x^2\) and \(x^7\), respectively. Using the rule \(\frac{x^a}{x^b} = x^{a - b}\):
   \[
   \frac{x^2}{x^7} = x^{2 - 7} = x^{-5}
   \]
3. **Variable \(y\)**: The powers of \(y\) in the numerator and denominator are \(y^5\) and \(y^2\), respectively:
   \[
   \frac{y^5}{y^2} = y^{5 - 2} = y^3
   \]

So, the expression inside the parentheses simplifies to:

\[
\frac{-3x^2 y^5}{2y^2 x^7} = -\frac{3}{2} x^{-5} y^3
\]

### Step 2: Apply the Exponent Outside the Parentheses
Now we have:

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

Using the property \((a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n\), we raise each part to the power of 3:

1. **Coefficient**: \(\left(-\frac{3}{2}\right)^3 = -\frac{27}{8}\)
2. **Variable \(x\)**: \((x^{-5})^3 = x^{-15}\)
3. **Variable \(y\)**: \((y^3)^3 = y^9\)

Combining these, we get:

\[
-\frac{27}{8} x^{-15} y^9
\]

### Final Answer

\[
-\frac{27}{8} x^{-15} y^9
\]

Would you like further clarification on any step, or do you have additional questions?

Here are some related questions to explore:

1. How do you simplify expressions with negative exponents?
2. What is the rule for raising a fraction to a power?
3. How do we handle exponents when variables are in both the numerator and denominator?
4. What happens to the sign when an expression with a negative base is raised to an odd power?
5. How does simplifying complex fractions with variables differ from working with only numbers?

**Tip**: Remember, when raising an entire fraction to an exponent, apply the exponent to each part of the fraction separately for easier calculations.

Simplify the expression \( \left( \frac{-3x^2 y^5}{2y^2 x^7} \right)^3 \).

Learn how to simplify the expression \( \left( \frac{-3x^2 y^5}{2y^2 x^7} \right)^3 \) using exponentiation rules. This problem involves simplifying algebraic fractions and applying laws of exponents, making it suitable for students in Grades 10-12.

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