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Let's go through this problem step-by-step. The question is asking to express \( 3 \cdot 9^{2x} \) in base 3.

### Step-by-Step Solution

1. **Rewrite 9 in terms of base 3:**  
   We know that \( 9 = 3^2 \). Therefore, we can rewrite \( 9^{2x} \) as:
   \[
   9^{2x} = (3^2)^{2x}
   \]

2. **Apply the power rule:**  
   Using the rule \( (a^m)^n = a^{m \cdot n} \), we get:
   \[
   (3^2)^{2x} = 3^{4x}
   \]

3. **Substitute back:**  
   Now we substitute \( 9^{2x} \) with \( 3^{4x} \):
   \[
   3 \cdot 9^{2x} = 3 \cdot 3^{4x}
   \]

4. **Combine the terms:**  
   Since \( 3 = 3^1 \), we can rewrite the expression as:
   \[
   3 \cdot 3^{4x} = 3^{1 + 4x}
   \]

### Final Answer
The expression \( 3 \cdot 9^{2x} \) in base 3 is:
\[
3^{1 + 4x}
\]

Would you like further details on any step, or have any questions?

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Here are 5 related questions that expand on this concept:

1. How would you express \( 3 \cdot 27^{x} \) in base 3?
2. If given \( 3^5 \cdot 9^{x+1} \), how would you simplify it in base 3?
3. What general rule can we use for converting expressions with \( 9^n \) or \( 27^n \) to base 3?
4. How would you solve \( 3^{1+4x} = 81 \) for \( x \)?
5. What other bases besides 3 make this kind of simplification easier?

**Tip:** When simplifying expressions involving exponents, always look for opportunities to rewrite terms in a common base. This allows you to apply exponent rules more effectively.

Step-by-step solution to convert the expression 3 * 9^(2x) to base 3. This problem involves understanding exponents and applying the power rule. Ideal for students in Grades 9-11.

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