To simplify the expression $\frac{9x^3}{3x^{-3}}$, follow these steps:

1. Simplify the coefficients: $\frac{9}{3} = 3$

2. Simplify the exponents of $x$ using the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$: $\frac{x^3}{x^{-3}} = x^{3 - (-3)} = x^{3 + 3} = x^6$

3. Combine the results: $3x^6$

So, the simplified expression is $\boxed{3x^6}$.

Would you like more details or have any questions about this solution? Here are some related questions to consider:

1. How do you simplify expressions involving negative exponents?
2. What is the quotient rule for exponents, and how is it applied?
3. Can you explain the product of powers rule with an example?
4. How would the solution change if the base was a different variable, say $y$, instead of $x$?
5. How can you apply this method to simplify the expression $\frac{4x^5}{2x^{-2}}$?

Tip: Remember, subtracting a negative exponent is the same as adding its positive counterpart!