rewrite the exponential expression to have the indicated base. 9^2x, base 3

To rewrite the exponential expression \(9^{2x}\) with base 3, follow these steps:

1. **Express 9 as a power of 3**: 
   \[
   9 = 3^2
   \]

2. **Substitute this into the original expression**:
   \[
   9^{2x} = (3^2)^{2x}
   \]

3. **Apply the power rule** \( (a^m)^n = a^{mn} \):
   \[
   (3^2)^{2x} = 3^{4x}
   \]

Thus, the expression \(9^{2x}\) rewritten with base 3 is \(3^{4x}\).

Would you like more details on this process, or do you have any questions?

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**Related questions:**

1. How would you rewrite \(27^x\) with base 3?
2. Can you express \(81^y\) using base 3?
3. How do you convert \(3^{6x}\) back to a base of 9?
4. How would you express \(16^x\) with base 2?
5. What is the equivalent of \(32^{3y}\) with base 2?
6. How would you simplify \((9^x) \times (9^{2x})\) using base 3?
7. How do you rewrite \(25^{4x}\) using base 5?
8. What is the general rule for converting exponential expressions to a different base?

**Tip:** When converting exponential expressions to different bases, always express the original base as a power of the new base, and then apply the power rule.

Learn how to rewrite the exponential expression 9^2x with base 3 using the power rule for exponents. This mathematical problem involves converting exponential functions to different bases, crucial for understanding algebraic manipulations. Suitable for Grades 9-12.

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