`{:("Column A",0 < x < y,"Column B"),(7^(1/x - 1/y),,7^(x - y)):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the values in Column A and Column B under the condition \(0 < x < y\).

### Step-by-Step Solution:

1. **Identify the expressions:**
- Column A: \(7^{\left(\frac{1}{x} - \frac{1}{y}\right)}\)
- Column B: \(7^{(x - y)}\)

2. **Understand the condition:**
- We are given that \(0 < x < y\). This implies that both \(x\) and \(y\) are positive numbers and \(y - x > 0\).

3. **Simplify Column A:**
- We can rewrite the expression in Column A:
\[ \frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy} \]
- Therefore, Column A becomes:
\[ 7^{\left(\frac{y - x}{xy}\right)} \]
- Since \(y - x > 0\) and \(xy > 0\) (as both \(x\) and \(y\) are positive), we conclude that:
\[ \frac{y - x}{xy} > 0 \]
- Thus, Column A simplifies to:
\[ 7^{\text{(positive number)}} \]

4. **Simplify Column B:**
- The expression in Column B is:
\[ 7^{(x - y)} \]
- Since \(y > x\), it follows that \(x - y < 0\). Therefore, Column B can be expressed as:
\[ 7^{(x - y)} = 7^{\text{(negative number)}} \]
- By the properties of exponents, we know that:
\[ 7^{\text{(negative number)}} = \frac{1}{7^{\text{(positive number)}}} \]

5. **Comparison of Column A and Column B:**
- Now we have:
- Column A: \(7^{\text{(positive number)}}\)
- Column B: \(\frac{1}{7^{\text{(positive number)}}}\)
- Since \(7^{\text{(positive number)}} > 1\), it follows that:
\[ 7^{\text{(positive number)}} > \frac{1}{7^{\text{(positive number)}}} \]
- Therefore, we conclude that:
\[ \text{Column A} > \text{Column B} \]

6. **Final Conclusion:**
- Based on our analysis, Column A is larger than Column B. Thus, the answer is:
- **Option A: Column A is larger.**