Let's solve the equation step by step:

You have the equation:

\[
\frac{3^{2x}}{3^4} = 3^{6x} \times 3^{10}
\]

### Step 1: Simplify the left-hand side (LHS)
Using the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\), the left-hand side becomes:

\[
\frac{3^{2x}}{3^4} = 3^{2x - 4}
\]

### Step 2: Simplify the right-hand side (RHS)
Using the rule \(a^m \times a^n = a^{m+n}\), the right-hand side becomes:

\[
3^{6x} \times 3^{10} = 3^{6x + 10}
\]

Now the equation is:

\[
3^{2x - 4} = 3^{6x + 10}
\]

### Step 3: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[
2x - 4 = 6x + 10
\]

### Step 4: Solve for \(x\)
Now solve for \(x\):

\[
2x - 4 = 6x + 10
\]

Subtract \(2x\) from both sides:

\[
-4 = 4x + 10
\]

Subtract 10 from both sides:

\[
-14 = 4x
\]

Divide by 4:

\[
x = -\frac{14}{4} = -\frac{7}{2}
\]

### Final Answer:
\[
x = -\frac{7}{2}
\]

Would you like more details or further clarifications? 

Here are 5 related questions you might explore:
1. How can you apply other exponent rules to simplify expressions like this?
2. How do you solve equations where exponents involve multiple variables?
3. What happens if the bases are different but the exponents are the same?
4. How do logarithms help in solving equations with exponents?
5. Can this problem be solved graphically to visualize the solution?

**Tip**: When working with exponent equations, always check if the bases are the same before comparing exponents.

Explore how to solve the exponent equation 3^2x / 3^4 = 3^6x * 3^10 using exponent rules. This detailed solution demonstrates step-by-step techniques for simplifying exponent expressions and solving algebraic equations. Suitable for students in Grades 9-12.

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