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.
`h^(4)k^(3)m^(2)`, given that `klt0 and hm ne0`

Answer

To determine whether the expression \( h^4 k^3 m^2 \) is positive, negative, or cannot be determined, we will analyze each component of the expression based on the given conditions.

### Step 1: Analyze \( h^4 \)
- The exponent of \( h \) is 4, which is an even number.
- Any number raised to an even power is always non-negative (either positive or zero).
- Therefore, \( h^4 \) is **positive** (assuming \( h \neq 0 \)).

**Hint:** Remember that even powers yield non-negative results.

### Step 2: Analyze \( k^3 \)
- The exponent of \( k \) is 3, which is an odd number.
- The sign of an odd power is the same as the sign of the base.
- We are given that \( k < 0 \), which means \( k \) is negative.
- Therefore, \( k^3 \) is **negative**.

**Hint:** Odd powers retain the sign of the base.

### Step 3: Analyze \( m^2 \)
- The exponent of \( m \) is 2, which is also an even number.
- Similar to \( h^4 \), any number raised to an even power is non-negative.
- Therefore, \( m^2 \) is **positive** (assuming \( m \neq 0 \)).

**Hint:** Again, even powers yield non-negative results.

### Step 4: Combine the results
Now, we combine the results of each component:
- \( h^4 \) is positive.
- \( k^3 \) is negative.
- \( m^2 \) is positive.

The expression can be simplified as:
\[ h^4 k^3 m^2 = (\text{positive}) \times (\text{negative}) \times (\text{positive}) \]

### Step 5: Determine the overall sign
- When multiplying a positive number with a negative number, the result is negative.
- The product of the positive \( h^4 \) and \( m^2 \) with the negative \( k^3 \) results in:
\[ \text{positive} \times \text{negative} = \text{negative} \]

Thus, the overall expression \( h^4 k^3 m^2 \) is **negative**.

### Final Answer
The expression \( h^4 k^3 m^2 \) is **negative**.

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