To solve the equation \( 9^x = \frac{1}{729} \), follow these steps:

1. **Express both sides as powers of the same base**:
   - \( 9 \) can be written as \( 3^2 \).
   - \( 729 \) can be written as \( 3^6 \) because \( 729 = 3^6 \).

   So the equation becomes:
   \[
   (3^2)^x = \frac{1}{3^6}
   \]

2. **Simplify the equation**:
   \[
   3^{2x} = 3^{-6}
   \]
   (The negative exponent indicates the reciprocal.)

3. **Set the exponents equal**:
   Since the bases are the same, the exponents must be equal:
   \[
   2x = -6
   \]

4. **Solve for \( x \)**:
   \[
   x = \frac{-6}{2} = -3
   \]

So, the solution is \( x = -3 \).

Would you like more details or have any questions about the steps? Here are some related questions you might find helpful:

1. How do you solve exponential equations with different bases?
2. What is the process of converting an equation into logarithmic form?
3. How can you solve equations where the base is not an integer?
4. What are the properties of exponents that are most useful in solving equations?
5. How does changing the base of an exponent affect the solution?

**Tip:** Always try to express both sides of an exponential equation with the same base to simplify the solution process.

Learn how to solve the exponential equation 9^x = 1/729 step-by-step with detailed explanations. This problem involves understanding exponential equations and powers of numbers. Suitable for students in Grades 9-12.

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