To simplify the given expression:

$\frac{3a \cdot 6a^7}{a^{-4}}$

Step-by-Step Simplification:

1. Multiply the terms in the numerator: $3a \cdot 6a^7 = 18a^{1+7} = 18a^8$

2. Simplify the division by subtracting the exponents of $a$: $\frac{18a^8}{a^{-4}} = 18a^{8 - (-4)} = 18a^{8 + 4} = 18a^{12}$

Final Answer:

$18a^{12}$

Let me know if you'd like additional details or have any questions!

Here are 5 related questions to explore this topic further:

1. How do you simplify expressions involving variables with multiple exponents?
2. What is the rule for multiplying powers with the same base?
3. What is the rule for dividing powers with the same base?
4. Can negative exponents ever lead to fractions in simplified results?
5. How do coefficients (e.g., numbers like 3 and 6) impact the calculation?

Tip: When dealing with exponents, remember that adding and subtracting exponents only works when the base is the same!