The equation in the image is:

$3^{2x+1} = 3^{2x} + 162$

Steps to solve for $x$:

1. Start by rewriting the left side: $3^{2x+1} = 3^{2x} \cdot 3$ So, the equation becomes: $3^{2x} \cdot 3 = 3^{2x} + 162$

2. Let $y = 3^{2x}$, to simplify the equation: $3y = y + 162$

3. Rearrange the equation: $3y - y = 162$ Simplifying: $2y = 162$ Divide both sides by 2: $y = 81$

4. Since $y = 3^{2x}$, we now have: $3^{2x} = 81$ Rewrite 81 as $3^4$: $3^{2x} = 3^4$

5. Equate the exponents: $2x = 4$

6. Solve for $x$: $x = \frac{4}{2} = 2$

Thus, the value of $x$ is:

$\boxed{2}$

Would you like more details on any of the steps?

Related questions:

1. How can we solve similar equations with different bases?
2. What are the key properties of exponents that help in solving these types of equations?
3. How would the solution change if the equation involved logarithms instead of exponents?
4. What are some other methods to solve exponential equations when the exponents are not directly comparable?
5. How can you apply these principles to solve equations involving both exponential and linear terms?

Tip:

When solving exponential equations, always look for ways to express terms with a common base to simplify comparisons between exponents.