Let's solve the given expression step by step.

We are tasked with simplifying the expression:

$\frac{(5ab)^3}{30a^{-6}b^{-7}}$

Step 1: Simplify the numerator

The numerator is $(5ab)^3$. Using the power of a product rule, we can distribute the exponent:

$(5ab)^3 = 5^3 \cdot a^3 \cdot b^3 = 125a^3b^3$

So, the numerator simplifies to:

$125a^3b^3$

Step 2: Simplify the denominator

The denominator is $30a^{-6}b^{-7}$, which can be written as:

$30a^{-6}b^{-7}$

Step 3: Combine the expression

We now have:

$\frac{125a^3b^3}{30a^{-6}b^{-7}}$

Step 4: Simplify the fraction

First, simplify the numerical coefficient:

$\frac{125}{30} = \frac{25}{6}$

Now simplify the powers of $a$ and $b$ using the exponent rules. For the powers of $a$, we have:

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

For the powers of $b$, we have:

$\frac{b^3}{b^{-7}} = b^{3 - (-7)} = b^{3 + 7} = b^{10}$

Step 5: Final expression

After combining everything, we get:

$\frac{25a^9b^{10}}{6}$

Thus, the simplified expression is:

$\boxed{\frac{25a^9b^{10}}{6}}$

This matches the fourth option.

Would you like further details or clarifications?

Here are some related questions you might find useful:

1. How do exponent rules apply to simplify complex fractions?
2. Can you explain the power of a product rule in detail?
3. How can negative exponents be handled when simplifying expressions?
4. What are some common mistakes to avoid when simplifying exponents?
5. How can you simplify expressions with multiple variables and exponents efficiently?

Tip: When simplifying expressions with exponents, always keep track of the signs of the exponents to avoid errors, especially when dealing with negative exponents.