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.