Define a @ b to be `a^3 - b`
`{:("Column A" , " ","ColumnB"),(x " @ "1, ,x " @ " 10):}`

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 evaluate the expressions defined in Column A and Column B using the operation defined as \( a @ b = a^3 - b \).

### Step-by-Step Solution:

1. **Define the Operation**:
We have the operation defined as:
\[ a @ b = a^3 - b \]

2. **Evaluate Column A**:
For Column A, we need to evaluate \( x @ 1 \):
\[ x @ 1 = x^3 - 1 \]

3. **Evaluate Column B**:
For Column B, we need to evaluate \( x @ 10 \):
\[ x @ 10 = x^3 - 10 \]

4. **Compare Column A and Column B**:
We need to compare \( x^3 - 1 \) (Column A) and \( x^3 - 10 \) (Column B):
\[ x^3 - 1 \quad \text{and} \quad x^3 - 10 \]
To compare these, we can rearrange the inequality:
\[ x^3 - 1 > x^3 - 10 \]
Simplifying this gives:
\[ -1 > -10 \]
This inequality is always true, regardless of the value of \( x \).

5. **Conclusion**:
Since \( x^3 - 1 \) is always greater than \( x^3 - 10 \), we conclude that:
\[ \text{Column A is greater than Column B for all } x. \]

### Final Answer:
The correct option is **Option 1: Column A is larger**.