The equation given in the image is:

$3^{3x - 2} = \frac{1}{81^{x + 1}}$

To solve this equation, let's follow these steps:

1. Rewrite the right side using powers of 3: Since $81 = 3^4$, we can rewrite $81^{x + 1}$ as $(3^4)^{x + 1}$. This simplifies to:

   $81^{x + 1} = 3^{4(x + 1)}$

   So, the equation now becomes:

   $3^{3x - 2} = \frac{1}{3^{4(x + 1)}}$

2. Rewrite the fraction as a negative exponent: Using the property $\frac{1}{a^b} = a^{-b}$, we can rewrite the right side:

   $\frac{1}{3^{4(x + 1)}} = 3^{-4(x + 1)}$

   Therefore, the equation simplifies to:

   $3^{3x - 2} = 3^{-4(x + 1)}$

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

   $3x - 2 = -4(x + 1)$

4. Solve the linear equation: Expand and simplify:

   $3x - 2 = -4x - 4$

   Combine like terms:

   $3x + 4x = -4 + 2$

   $7x = -2$

   $x = -\frac{2}{7}$

Thus, the solution is:

$\boxed{x = -\frac{2}{7}}$

Would you like further details or explanations on this solution?

Questions to Explore Further:

1. What are other methods to solve exponential equations?
2. How do you rewrite terms using similar bases in exponent problems?
3. What are the common properties of exponents used in solving equations?
4. How can we verify the solution of an exponential equation?
5. What happens if the bases in an exponential equation are different?

Tip:

Always try to express terms with the same base when solving exponential equations; this makes it easier to set exponents equal and solve.