We are tasked with simplifying the expression $(3a b^{-3})^3 \times (2a^4)^{-2}$ into a form with positive exponents. Let's break it down step by step:

Given Expression:

$(3a b^{-3})^3 \times (2a^4)^{-2}$

Step 1: Apply exponent rules

For the first part, $(3a b^{-3})^3$, distribute the exponent of 3 across each term: $(3a b^{-3})^3 = 3^3 \cdot a^3 \cdot b^{-9} = 27a^3 b^{-9}$

For the second part, $(2a^4)^{-2}$, apply the exponent of $-2$ across both terms: $(2a^4)^{-2} = 2^{-2} \cdot (a^4)^{-2} = 2^{-2} \cdot a^{-8} = \frac{1}{4a^8}$

Step 2: Multiply the two results:

Now, multiply the two expressions: $27a^3 b^{-9} \times \frac{1}{4a^8} = \frac{27a^3 b^{-9}}{4a^8}$

Step 3: Simplify the exponents:

For the $a$-terms: $a^3 / a^8 = a^{-5}$

Thus, we have: $\frac{27 b^{-9}}{4a^5} = \frac{27}{4a^5 b^9}$

Final Expression:

The simplified form is $\frac{27}{4a^5 b^9}$.

Answer:

From the options provided in the image, the correct answer is C: $\frac{3^3}{2^2 \cdot a^5 \cdot b^9}$

Would you like further details or explanations on any part of this process?

Here are five related questions to expand your understanding:

1. How do we apply exponent rules when there are negative exponents?
2. Why do we multiply exponents when raising a power to another power?
3. What happens when multiplying terms with the same base but different exponents?
4. How do negative exponents affect the position of terms in a fraction?
5. How do we handle fractional exponents during simplification?

Tip: Always remember that a negative exponent means the reciprocal of the base raised to the positive exponent.