`{:("Column A"," ","Column B"),((((sqrt7)^(x))^(2))/((sqrt7)^(11)),,(7^(x))/(7^(13))):}`

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 expressions in Column A and Column B.

### Step 1: Write down the expressions from Column A and Column B.
- Column A: \(\frac{(\sqrt{7})^{x^2}}{(\sqrt{7})^{11}}\)
- Column B: \(\frac{7^x}{7^{13}}\)

### Step 2: Simplify the expression in Column A.
We know that \(\sqrt{7} = 7^{1/2}\). Therefore, we can rewrite the expression in Column A as:
\[ \frac{(7^{1/2})^{x^2}}{(7^{1/2})^{11}} \]

### Step 3: Apply the power of a power property.
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify further:
\[ \frac{7^{(1/2) \cdot x^2}}{7^{(1/2) \cdot 11}} = \frac{7^{x^2/2}}{7^{11/2}} \]

### Step 4: Use the quotient property of exponents.
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we can combine the powers:
\[ 7^{(x^2/2) - (11/2)} = 7^{(x^2 - 11)/2} \]

### Step 5: Simplify the expression in Column B.
The expression in Column B can be simplified using the same quotient property:
\[ \frac{7^x}{7^{13}} = 7^{x - 13} \]

### Step 6: Compare the two simplified expressions.
Now we have:
- Column A: \(7^{(x^2 - 11)/2}\)
- Column B: \(7^{(x - 13)}\)

To compare these two expressions, we need to analyze the exponents:
1. Column A exponent: \(\frac{x^2 - 11}{2}\)
2. Column B exponent: \(x - 13\)

### Step 7: Set up the inequality for comparison.
We need to determine when:
\[ \frac{x^2 - 11}{2} > x - 13 \]

### Step 8: Clear the fraction by multiplying both sides by 2.
\[ x^2 - 11 > 2(x - 13) \]
\[ x^2 - 11 > 2x - 26 \]

### Step 9: Rearrange the inequality.
\[ x^2 - 2x + 15 > 0 \]

### Step 10: Find the roots of the quadratic equation.
To find the roots, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -2\), and \(c = 15\):
\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 - 60}}{2} = \frac{2 \pm \sqrt{-56}}{2} \]
Since the discriminant is negative, the quadratic has no real roots and is always positive.

### Conclusion:
Since \(x^2 - 2x + 15 > 0\) for all real \(x\), it follows that Column A is always greater than Column B. Therefore, the correct option is that Column A is larger.

### Final Answer:
Column A is larger than Column B.

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