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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**:
\[
\frac{3^3}{2^2 \cdot a^5 \cdot b^9}
\]

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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.

Simplify the expression (3a b^{-3})^3 × (2a^4)^{-2} and write the result with positive exponents.

Learn how to simplify the expression (3a b^{-3})^3 × (2a^4)^{-2} step by step. This example covers key exponentiation rules, including multiplying powers and dealing with negative exponents. Suitable for students in Grades 9-11.

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