https://file.aiquickdraw.com/mathbot/file/1725514690003lkpfsbww.jpg

The equation in the image is:

\[
9^{3x + 2} = \frac{1}{3^{2x - 5}}
\]

To solve for \(x\), we can begin by expressing both sides with the same base. Notice that \(9\) can be written as \(3^2\), so the equation becomes:

\[
(3^2)^{3x + 2} = \frac{1}{3^{2x - 5}}
\]

Using the property of exponents \((a^m)^n = a^{m \cdot n}\), the left-hand side becomes:

\[
3^{2(3x + 2)} = 3^{- (2x - 5)}
\]

Simplifying both sides:

\[
3^{6x + 4} = 3^{-(2x - 5)}
\]

Now, since the bases are the same, we can equate the exponents:

\[
6x + 4 = -(2x - 5)
\]

Expanding the right-hand side:

\[
6x + 4 = -2x + 5
\]

Now, solve for \(x\):

\[
6x + 2x = 5 - 4
\]

\[
8x = 1
\]

\[
x = \frac{1}{8}
\]

So, the solution is:

\[
x = \frac{1}{8}
\]

Would you like more details, or do you have any questions?

Here are 5 follow-up questions you might explore:
1. How would you solve this equation if the bases were different?
2. Can you express \(x\) as a decimal?
3. What if the equation involved logarithms? How would that change the approach?
4. How do exponent rules apply when simplifying both sides of the equation?
5. Could there be any extraneous solutions if this were part of a system of equations?

**Tip**: Always check if you can express different numbers using the same base when solving exponential equations.

Solve the exponential equation 9^(3x + 2) = 1 / 3^(2x - 5)

Step-by-step solution for solving the exponential equation 9^(3x + 2) = 1 / 3^(2x - 5). This problem demonstrates how to express different numbers using the same base and solve for x using properties of exponents. Suitable for high school students (Grades 9-12).

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