To solve the equation \( 81^{x-33} = 3^{120-2x} \), we'll start by expressing both sides of the equation with the same base. 

### Step 1: Express \( 81 \) as a power of \( 3 \).
We know that:
\[
81 = 3^4
\]
So, the equation becomes:
\[
(3^4)^{x-33} = 3^{120-2x}
\]

### Step 2: Simplify the exponents.
Using the power of a power rule \( (a^m)^n = a^{mn} \), we can simplify the left-hand side:
\[
3^{4(x-33)} = 3^{120-2x}
\]

### Step 3: Set the exponents equal to each other.
Since the bases are the same (both are base 3), we can set the exponents equal to each other:
\[
4(x-33) = 120 - 2x
\]

### Step 4: Solve for \( x \).
Expand and solve the equation:
\[
4x - 132 = 120 - 2x
\]
Add \( 2x \) to both sides to get:
\[
6x - 132 = 120
\]
Add 132 to both sides:
\[
6x = 252
\]
Finally, divide by 6:
\[
x = 42
\]

### Final Answer:
The solution to the equation is \( \boxed{x = 42} \).

Would you like further details on any of the steps, or do you have any additional questions?

Here are five related questions you might find interesting:
1. How do you solve exponential equations with different bases?
2. What are the properties of exponents that are frequently used in solving equations?
3. How do logarithms help in solving exponential equations?
4. How can you solve exponential equations graphically?
5. What are some real-life applications of exponential equations?

**Tip:** When solving exponential equations, always try to express both sides with the same base to simplify the process.

Learn how to solve the exponential equation 81^(x-33) = 3^(120-2x) step-by-step. This problem involves exponential equations and the power of a power rule. Suitable for high school students in Grades 10-12.

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