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The problem in the image asks to simplify the expression \((3a^2b^{-3})^3 \times (2a)^{-2}\) into a form with positive exponents. Let's break it down step by step:

### Expression:
\[
(3a^2b^{-3})^3 \times (2a)^{-2}
\]

### Step 1: Simplify each term
- **First term: \((3a^2b^{-3})^3\)**  
  Apply the power rule to each factor inside the parentheses:
  \[
  (3a^2b^{-3})^3 = 3^3 \cdot (a^2)^3 \cdot (b^{-3})^3 = 27a^6b^{-9}
  \]

- **Second term: \((2a)^{-2}\)**  
  Apply the power to each factor:
  \[
  (2a)^{-2} = 2^{-2} \cdot a^{-2} = \frac{1}{2^2 \cdot a^2} = \frac{1}{4a^2}
  \]

### Step 2: Combine both terms
Now, multiply the results from both terms:
\[
27a^6b^{-9} \times \frac{1}{4a^2} = \frac{27a^6b^{-9}}{4a^2}
\]

### Step 3: Simplify the exponents
Use the exponent rules to simplify the powers of \(a\):
\[
\frac{27a^6}{4a^2} = \frac{27}{4}a^{6-2}b^{-9} = \frac{27}{4}a^4b^{-9}
\]

### Step 4: Write the expression with positive exponents
Since the exponent of \(b\) is negative, move it to the denominator:
\[
\frac{27a^4}{4b^9}
\]

Thus, the simplified form is:
\[
\frac{27a^4}{4b^9}
\]

### Conclusion:
The correct option is **C**, which matches \(\frac{27a^4}{4b^9}\).

---

Would you like further clarification on any step?

Here are five related questions to consider:
1. What are the general rules for multiplying exponents?
2. How do negative exponents affect a term's position in a fraction?
3. What happens when you raise a product of terms to a power?
4. Can you simplify \((a^m \cdot b^n)^p\)?
5. How do fractional exponents relate to roots?

**Tip:** Always apply the power rule individually to each factor within parentheses before simplifying.

Bentuk sederhana dari (3a^2b^{-3})^3 × (2a)^{-2} dalam bentuk eksponen positif adalah . . .

Learn how to simplify the expression (3a^2b^{-3})^3 × (2a)^{-2} into a form with positive exponents. This problem involves applying the power of a power rule, handling negative exponents, and simplifying algebraic expressions. Suitable for students in Grades 8-10.

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