To solve this problem, let's simplify the given expression step-by-step:

The expression is:

$\frac{-9x^{-1}y^{-9}}{-15x^5y^{-3}}$

Step 1: Simplify the Coefficients

First, we simplify the coefficients $\frac{-9}{-15}$:

$\frac{-9}{-15} = \frac{9}{15} = \frac{3}{5}$

Step 2: Apply the Exponent Rules

Now, let's simplify the powers of $x$ and $y$. Recall the rules of exponents:

• When dividing like bases, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Apply this rule to $x$ and $y$:

$\frac{x^{-1}}{x^5} = x^{-1 - 5} = x^{-6}$

$\frac{y^{-9}}{y^{-3}} = y^{-9 - (-3)} = y^{-9 + 3} = y^{-6}$

Step 3: Combine Results

Combining these results, the simplified expression is:

$\frac{3}{5x^6y^6}$

Conclusion

The answer that matches this simplified expression is:

$\boxed{\frac{3}{5x^6y^6}}$

This corresponds to the second option in the provided choices.

If you have any questions or need more details, let me know!

Related Questions:

1. How do you handle negative exponents in expressions?
2. What are the rules for multiplying and dividing exponents?
3. How do you simplify expressions with different bases?
4. What are some common mistakes when working with exponent rules?
5. How can you use exponent rules to solve equations?

Tip:

Always double-check your work by verifying each step, especially when applying exponent rules. This helps to avoid mistakes due to sign errors or incorrect arithmetic.