Decide whether the expression described is Positive, Negative, or Cannot Be Determined. If you answer Cannot Be Determined, give numerical examples to show how the problem could be either positive or negative.
`|x| xxy^(2)`, given that `xy ne0`

Answer

To determine whether the expression \( |x| \cdot y^2 \) is positive, negative, or cannot be determined, we can analyze each component of the expression step by step.

### Step 1: Analyze the modulus of \( x \)
The modulus function \( |x| \) gives the absolute value of \( x \). By definition, the absolute value is always non-negative. This means:
- If \( x > 0 \), then \( |x| = x \) (positive).
- If \( x < 0 \), then \( |x| = -x \) (also positive).
- If \( x = 0 \), then \( |x| = 0 \).

Thus, we can conclude that \( |x| \geq 0 \).

**Hint:** Remember that the modulus function always yields a non-negative result.

### Step 2: Analyze \( y^2 \)
Next, we consider \( y^2 \). The square of any real number (whether positive or negative) is always non-negative. Therefore:
- If \( y > 0 \), then \( y^2 > 0 \).
- If \( y < 0 \), then \( y^2 > 0 \).
- If \( y = 0 \), then \( y^2 = 0 \).

Thus, we can conclude that \( y^2 \geq 0 \).

**Hint:** Squaring any real number results in a non-negative value.

### Step 3: Combine the results
Now we combine the results from the first two steps:
- Since \( |x| \geq 0 \) and \( y^2 \geq 0 \), the product \( |x| \cdot y^2 \) will also be non-negative.
- Specifically, \( |x| \cdot y^2 \) will be positive if both \( |x| > 0 \) and \( y^2 > 0 \). It will be zero if either \( |x| = 0 \) or \( y^2 = 0 \).

Given the condition \( xy \neq 0 \), we know that neither \( x \) nor \( y \) can be zero. Therefore:
- \( |x| > 0 \) and \( y^2 > 0 \).

### Conclusion
Since both components of the expression are positive, we can conclude that:
\[ |x| \cdot y^2 > 0 \]
Thus, the expression \( |x| \cdot y^2 \) is **positive**.

**Final Answer:** The expression is positive.