We are given the equation:

\[
\frac{9^3}{9^x} = \frac{1}{9^6}
\]

### Step 1: Simplify the left-hand side of the equation
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify the left-hand side:

\[
\frac{9^3}{9^x} = 9^{3 - x}
\]

Thus, the equation becomes:

\[
9^{3 - x} = 9^{-6}
\]

### Step 2: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[
3 - x = -6
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\):

\[
3 - x = -6
\]

Subtract 3 from both sides:

\[
-x = -9
\]

Multiply both sides by \(-1\):

\[
x = 9
\]

### Final Answer:
\[
x = 9
\]

Would you like more details on any step?

Here are 5 related questions:
1. What are the laws of exponents used in simplifying expressions?
2. How do you solve exponential equations with different bases?
3. How do negative exponents affect fractions?
4. Can you apply logarithms to solve equations like this?
5. What is the general form for solving equations involving powers?

**Tip:** Always check if you can simplify an equation using the properties of exponents when dealing with powers.

Learn how to solve the exponential equation 9^3 / 9^x = 1 / 9^6 using exponent laws. This problem demonstrates the application of the division and negative exponent rules. Suitable for students in Grades 8-10.

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