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{:("Column A",x ge 1, "Column B"),(x^(10...

`{:("Column A",x ge 1, "Column B"),(x^(10),,x^(100)):}`

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

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to compare the values in Column A and Column B given the condition \( x \geq 1 \). ### Step-by-Step Solution: 1. **Identify the expressions in each column**: - Column A: \( x^{10} \) - Column B: \( x^{100} \) 2. **Evaluate the expressions at the boundary condition \( x = 1 \)**: - For Column A: \[ x^{10} = 1^{10} = 1 \] - For Column B: \[ x^{100} = 1^{100} = 1 \] - At \( x = 1 \), both Column A and Column B are equal to 1. 3. **Evaluate the expressions for \( x > 1 \)**: - Let's take \( x = 2 \) (a value greater than 1): - For Column A: \[ x^{10} = 2^{10} = 1024 \] - For Column B: \[ x^{100} = 2^{100} = 1.267 \times 10^{30} \quad (\text{a very large number}) \] - Here, \( 2^{100} \) is significantly larger than \( 2^{10} \). 4. **Generalize the comparison for \( x > 1 \)**: - For any \( x > 1 \), since \( 100 > 10 \), it follows that: \[ x^{100} > x^{10} \] - Thus, Column B will always be greater than Column A for any \( x > 1 \). 5. **Conclusion**: - At \( x = 1 \), both columns are equal. - For \( x > 1 \), Column B is greater than Column A. - Therefore, we can conclude that Column B is greater than or equal to Column A for \( x \geq 1 \). ### Final Answer: - The answer is that Column B is greater than Column A for \( x > 1 \), and they are equal when \( x = 1 \). Thus, the correct option is **D** (there is not enough information to decide).
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