For all negative even integers x , which of the following is a correct ordering of the terms `x,x^x , ((-x)!)^x`, and `((-x)!)^"(-x)!"` ?

Options

A. 1
B. `((-x)!)^"(-x)!" ge x^x ge ((-x)!)^x ge x`
C. 3
D. 4

Answer

To solve the problem of ordering the terms \( x, x^x, ((-x)!)^x, \) and \( ((-x)!)^{(-x)!} \) for all negative even integers \( x \), we will follow these steps:

### Step 1: Choose a value for \( x \)
We will substitute \( x = -4 \) (a negative even integer) into the terms.

### Step 2: Calculate each term
1. **Calculate \( x \)**:
\[ x = -4 \]

2. **Calculate \( x^x \)**:
\[ x^x = (-4)^{-4} = \frac{1}{(-4)^4} = \frac{1}{256} \]

3. **Calculate \( (-x)! \)**:
\[ -x = 4 \implies (-x)! = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

4. **Calculate \( ((-x)!)^x \)**:
\[ ((-x)!)^x = (4!)^{-4} = 24^{-4} = \frac{1}{24^4} \]
Now calculate \( 24^4 \):
\[ 24^4 = 331776 \implies ((-x)!)^x = \frac{1}{331776} \]

5. **Calculate \( ((-x)!)^{(-x)!} \)**:
\[ ((-x)!)^{(-x)!} = (4!)^{4!} = 24^{24} \]

### Step 3: Order the terms
Now we have the values of all terms:
- \( x = -4 \)
- \( x^x = \frac{1}{256} \)
- \( ((-x)!)^x = \frac{1}{331776} \)
- \( ((-x)!)^{(-x)!} = 24^{24} \)

### Step 4: Compare the values
1. **Identify the largest term**:
- \( 24^{24} \) is the largest because it is a positive power of a positive integer.

2. **Next largest**:
- \( \frac{1}{256} \) is greater than \( \frac{1}{331776} \) because \( 256 < 331776 \).

3. **Next**:
- \( \frac{1}{331776} \) is greater than \( -4 \).

### Final Ordering
Thus, the correct order from greatest to least is:
\[ (((-x)!)^{(-x)!}) > (x^x) > ((-x)!)^x > x \]
This translates to:
\[ 24^{24} > \frac{1}{256} > \frac{1}{331776} > -4 \]

### Conclusion
The correct ordering of the terms is:
\[ ((-x)!)^{(-x)!} > x^x > ((-x)!)^x > x \]