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We are given the equation \( 8^x = 9 \) and asked to find the value of \( 8^{-2x} \).

### Step-by-Step Solution:
1. Start with the given equation:
   \[
   8^x = 9
   \]

2. To solve for \( 8^{-2x} \), recall that:
   \[
   8^{-2x} = \left(8^x\right)^{-2}
   \]
   So, we substitute \( 8^x = 9 \):
   \[
   8^{-2x} = 9^{-2}
   \]

3. Now calculate \( 9^{-2} \):
   \[
   9^{-2} = \frac{1}{9^2} = \frac{1}{81}
   \]

Thus, \( 8^{-2x} = \frac{1}{81} \).

### Final Answer:
\[
8^{-2x} = \frac{1}{81}
\]

Let me know if you would like further details or have any questions!

Here are five related questions for practice:
1. If \( 8^x = 4 \), what is \( 8^{-x} \)?
2. Solve for \( x \) if \( 2^{3x} = 16 \).
3. What is the value of \( 27^{-x} \) if \( 3^x = 2 \)?
4. Simplify \( (8^x)^2 \) when \( 8^x = 5 \).
5. Find the value of \( 8^{x-1} \) if \( 8^x = 7 \).

**Tip:** When solving exponential equations, it's often helpful to express everything in terms of the same base if possible.

Learn how to solve the equation 8^x = 9 and find the value of 8^{-2x}. This detailed step-by-step guide will walk you through exponent rules and algebraic manipulations, including substituting values and simplifying expressions. Suitable for Grades 9-11.

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