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Let's analyze the pattern and equation options.

The given pattern of examples is:

1. \( \frac{6}{a} \# \frac{2}{a} = a^{14} \)
2. \( \frac{4}{a} \# \frac{3}{a} = a^{14} \)
3. \( \frac{5}{a} \# \frac{6}{a} = a^{32} \)

Our goal is to select the equation that describes the pattern.

### Examining the Examples:
- In the first example, the exponents 6 and 2 combine to form the exponent 14 in the result \(a^{14}\).
- In the second example, exponents 4 and 3 combine to form the same result \(a^{14}\).
- In the third example, the exponents 5 and 6 combine to form \(a^{32}\).

### Generalizing the Pattern:
We can assume that the operation \( \# \) takes two exponents and applies some rule to generate the final exponent.

### Testing the Equation Options:
Let’s consider each option:

#### Option A:
\[
\frac{x}{a} \# \frac{y}{a} = a^{2x + y}
\]
For the first example:
\[
\frac{6}{a} \# \frac{2}{a} = a^{2(6) + 2} = a^{12 + 2} = a^{14}
\]
This holds true.

For the second example:
\[
\frac{4}{a} \# \frac{3}{a} = a^{2(4) + 3} = a^{8 + 3} = a^{11}
\]
This does not hold true, so this is incorrect.

#### Option B:
\[
\frac{x}{a} \# \frac{y}{a} = a^{x + 2y}
\]
For the first example:
\[
\frac{6}{a} \# \frac{2}{a} = a^{6 + 2(2)} = a^{6 + 4} = a^{10}
\]
This does not hold true.

#### Option C:
\[
\frac{x}{a} \# \frac{y}{a} = a^{x \cdot y + 2}
\]
For the first example:
\[
\frac{6}{a} \# \frac{2}{a} = a^{6 \cdot 2 + 2} = a^{12 + 2} = a^{14}
\]
This holds true.

For the second example:
\[
\frac{4}{a} \# \frac{3}{a} = a^{4 \cdot 3 + 2} = a^{12 + 2} = a^{14}
\]
This holds true as well.

For the third example:
\[
\frac{5}{a} \# \frac{6}{a} = a^{5 \cdot 6 + 2} = a^{30 + 2} = a^{32}
\]
This also holds true.

#### Option D:
\[
\frac{x}{a} \# \frac{y}{a} = a^{x + y + 4}
\]
For the first example:
\[
\frac{6}{a} \# \frac{2}{a} = a^{6 + 2 + 4} = a^{12}
\]
This does not hold true.

### Conclusion:
The correct equation is **Option C**, \( \frac{x}{a} \# \frac{y}{a} = a^{x \cdot y + 2} \).

Do you have any questions, or would you like further clarification?

Here are five follow-up questions to expand on this topic:

1. How can you test different functions to see if they fit a pattern in math problems?
2. What would happen if you changed the operation from multiplication to addition inside the exponent rule in Option C?
3. Can you create a similar pattern with different base values instead of 'a'?
4. What kind of real-world problems could involve exponent patterns like these?
5. How would the solution change if the exponents were negative instead of positive?

**Tip:** When analyzing patterns, breaking down each example step-by-step helps reveal the underlying mathematical rule.

Study the pattern in these examples and select the correct equation that describes it.

Explore how to analyze a series of examples with exponents and select the correct equation that describes the pattern. This problem focuses on exponent rules and algebraic operations, ideal for students in Grades 9-12.

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