The extracted equation appears to have some errors, likely due to text recognition issues. From the image, the correct equation seems to be:

$3^{2x} \cdot 9^{1-x} = 27$

Let's solve for $x$.

Step 1: Express $9$ and $27$ as powers of 3.

We know that:

• $9 = 3^2$
• $27 = 3^3$

So, the equation becomes: $3^{2x} \cdot (3^2)^{1-x} = 3^3$

Step 2: Simplify the exponents.

Using the power of a power rule, $(a^m)^n = a^{mn}$, we rewrite the equation as: $3^{2x} \cdot 3^{2(1-x)} = 3^3$

Now, combine the exponents on the left side: $3^{2x + 2(1-x)} = 3^3$

Step 3: Solve for the exponents.

Equate the exponents of 3: $2x + 2(1-x) = 3$

Expand and simplify: $2x + 2 - 2x = 3$

This simplifies to: $2 = 3$

Step 4: Interpret the result.

This is a contradiction, meaning there is no solution for $x$ in this equation.

Let me know if you need further clarification! Here are five related questions you might want to explore:

1. How do you solve exponential equations using logarithms?
2. Can an exponential equation have no solution, and why?
3. How do you solve equations with different bases?
4. What is the significance of contradictions in solving equations?
5. Can the equation $a^x = b^y$ have multiple solutions depending on values of $a$ and $b$?

Tip: Always verify the bases in exponential equations to simplify them efficiently!