Find all solutions $n$ to the equation
$$6^{2n+1}+6^2=6^n+6^{n+3}.$$

We are given the equation:

\[
6^{2n+1} + 6^2 = 6^n + 6^{n+3}.
\]

### Step 1: Simplify the equation
First, express $ 6^{n+3} $ as:

\[
6^{n+3} = 6^n \cdot 6^3 = 6^n \cdot 216.
\]

Substituting this into the equation, we get:

\[
6^{2n+1} + 6^2 = 6^n + 6^n \cdot 216.
\]

Now rewrite the equation:

\[
6^{2n+1} + 36 = 6^n + 216 \cdot 6^n.
\]

### Step 2: Factor the terms on the right-hand side
Factor out $ 6^n $ from the right-hand side:

\[
6^{2n+1} + 36 = 6^n(1 + 216).
\]

Simplify the right-hand side:

\[
6^{2n+1} + 36 = 6^n \cdot 217.
\]

### Step 3: Rewrite $ 6^{2n+1} $
We can rewrite $ 6^{2n+1} $ as:

\[
6^{2n+1} = 6 \cdot 6^{2n} = 6 \cdot (6^n)^2.
\]

Substitute this into the equation:

\[
6 \cdot (6^n)^2 + 36 = 6^n \cdot 217.
\]

### Step 4: Divide by $ 6^n $
Assume $ 6^n \neq 0 $, and divide both sides by $ 6^n $:

\[
6 \cdot 6^n + \frac{36}{6^n} = 217.
\]

This simplifies to:

\[
6 \cdot 6^n + 6^{-n} \cdot 36 = 217.
\]

### Step 5: Solve for specific values of $ n $

#### Case 1: $ n = 1 $
Substitute $ n = 1 $ into the equation:

\[
6 \cdot 6^1 + 6^{-1} \cdot 36 = 217.
\]

This gives:

\[
6 \cdot 6 + \frac{36}{6} = 217,
\]
\[
36 + 6 = 217,
\]
\[
42 \neq 217.
\]

Thus, $ n = 1 $ is not a solution.

#### Case 2: $ n = 2 $
Substitute $ n = 2 $ into the equation:

\[
6 \cdot 6^2 + 6^{-2} \cdot 36 = 217.
\]

This gives:

\[
6 \cdot 36 + \frac{36}{36} = 217,
\]
\[
216 + 1 = 217,
\]
\[
217 = 217.
\]

Thus, $ n = 2 $ is a solution.

#### Case 3: $ n = 0 $
Substitute $ n = 0 $ into the equation:

\[
6 \cdot 6^0 + 6^{-0} \cdot 36 = 217.
\]

This gives:

\[
6 \cdot 1 + 36 = 217,
\]
\[
6 + 36 = 217,
\]
\[
42 \neq 217.
\]

Thus, $ n = 0 $ is not a solution.

### Final Answer:
The only solution is:

\[
n = 2.
\]

Let me know if you'd like further details or have any questions! Here are 5 questions that could extend this concept:

1. How would the equation change if the base were different from 6?
2. Can this method be used to solve other exponential equations?
3. What happens if you have different powers on both sides of the equation?
4. How does the factorization technique help in solving exponential equations?
5. Can this equation have complex number solutions?

**Tip:** Always check small values of variables like $ n = 0, 1, 2 $ when dealing with exponential equations, as they often yield quick solutions.

Find all solutions $n$ to the equation $$6^{2n+1}+6^2=6^n+6^{n+3}.$$

Discover the solution to the exponential equation 6^(2n+1) + 6^2 = 6^n + 6^(n+3). This detailed step-by-step guide covers key algebraic concepts and properties of exponents. Suitable for students in Grades 10-12.

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