If `u^v + v^w + w^x = 0` For every negative integer u, v, w, x the value of `u xx v xx w xx x` is necessarily be :

Options

A. A. 0
B. B. less than zero
C. C. even
D. D. odd

Answer

To solve the problem, we need to analyze the equation \( u^v + v^w + w^x = 0 \) under the condition that \( u, v, w, x \) are all negative integers.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( u^v + v^w + w^x = 0 \) implies that the sum of three terms is zero. Since \( u, v, w, x \) are negative integers, we need to consider the nature of powers of negative integers.

**Hint**: Remember that the sign of a power depends on whether the exponent is odd or even.

2. **Analyzing Powers**:
- If \( u \) is a negative integer and \( v \) is also a negative integer, then \( u^v \) will be positive if \( v \) is even and negative if \( v \) is odd.
- Similarly, \( v^w \) and \( w^x \) will follow the same rule based on whether \( w \) and \( x \) are odd or even.

**Hint**: Check the parity (odd or even) of the exponents to determine the sign of each term.

3. **Finding Conditions for Zero**:
For the sum \( u^v + v^w + w^x = 0 \) to hold, one of the following must be true:
- All three terms are zero.
- One term is positive and the other two are negative such that they cancel out.

**Hint**: Consider the case where all terms are zero, which is a simpler scenario to analyze.

4. **Setting Values**:
Let's set \( u = -1 \), \( v = -2 \), \( w = -3 \), and \( x = -4 \) as an example:
- \( u^v = (-1)^{-2} = 1 \)
- \( v^w = (-2)^{-3} = -\frac{1}{8} \)
- \( w^x = (-3)^{-4} = \frac{1}{81} \)

Here, \( 1 - \frac{1}{8} + \frac{1}{81} \) does not equal zero, so we need to adjust our integers.

**Hint**: Try different combinations of negative integers to see if they can satisfy the equation.

5. **Conclusion on Product**:
The product \( u \times v \times w \times x \) will always yield a positive result since the product of four negative integers is positive.

**Final Calculation**:
- The product \( u \times v \times w \times x \) will be positive regardless of the specific negative integers chosen.

### Final Answer:
The value of \( u \times v \times w \times x \) is necessarily a positive integer.