https://file.aiquickdraw.com/mathbot/file/1730650470866k5g2x41i.jpg

To solve the equation:

\[
3^{x-2} + \frac{4}{3^{2 - x}} = 45
\]

we'll proceed as follows:

1. Rewrite \( \frac{4}{3^{2 - x}} \) by using properties of exponents:
   \[
   \frac{4}{3^{2 - x}} = 4 \cdot 3^{x - 2}
   \]

   Thus, the equation becomes:
   \[
   3^{x-2} + 4 \cdot 3^{x-2} = 45
   \]

2. Factor out \( 3^{x-2} \):
   \[
   3^{x-2} (1 + 4) = 45
   \]

3. Simplify inside the parentheses:
   \[
   3^{x-2} \cdot 5 = 45
   \]

4. Divide both sides by 5:
   \[
   3^{x-2} = 9
   \]

5. Recognize that \( 9 \) can be written as \( 3^2 \), so we rewrite the equation as:
   \[
   3^{x-2} = 3^2
   \]

6. Since the bases are the same, we can equate the exponents:
   \[
   x - 2 = 2
   \]

7. Solve for \( x \):
   \[
   x = 4
   \]

### Solution:
The value of \( x \) is \( 4 \).

Would you like further details or explanations?

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**Related Questions:**
1. What are other methods to solve exponential equations?
2. How do properties of exponents help in solving equations?
3. Can this problem be solved graphically, and how?
4. How do we handle equations with different bases in exponents?
5. What happens if the base is not a whole number?

**Tip:** When you see terms with the same base in exponential form, try factoring to simplify the equation.

Solve for x in the equation 3^(x-2) + 4 / 3^(2-x) = 45.

Discover the steps to solve the exponential equation 3^(x-2) + 4 / 3^(2-x) = 45. This solution demonstrates properties of exponents and simplification techniques, suitable for students in Grades 10-12.

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