If r is negative, which of the following must also be negative?

Options

A. `r^2`
B. `r^3`
C. `r^4`
D. `r^2 cdot r^3`

Answer

To solve the question, we need to determine which expressions must be negative if \( r \) is negative.

### Step-by-Step Solution:

1. **Identify the nature of \( r \)**:
- Given that \( r \) is negative, we can denote it as \( r < 0 \).

2. **Evaluate the first option: \( r^2 \)**:
- Since \( r^2 \) is an even power of \( r \), we calculate:
\[ r^2 = r \times r \]
A negative number multiplied by itself results in a positive number. Therefore, \( r^2 > 0 \).

3. **Evaluate the second option: \( r^3 \)**:
- Here, \( r^3 \) is an odd power of \( r \):
\[ r^3 = r \times r \times r \]
A negative number multiplied by itself an odd number of times remains negative. Thus, \( r^3 < 0 \).

4. **Evaluate the third option: \( r^4 \)**:
- \( r^4 \) is again an even power:
\[ r^4 = r \times r \times r \times r \]
Similar to \( r^2 \), this results in a positive number. Therefore, \( r^4 > 0 \).

5. **Evaluate the fourth option: \( r^2 \times r^3 \)**:
- We can rewrite this using the properties of exponents:
\[ r^2 \times r^3 = r^{2+3} = r^5 \]
Since \( r^5 \) is an odd power of a negative number, it will also be negative. Thus, \( r^5 < 0 \).

### Conclusion:
From the evaluations:
- \( r^2 > 0 \) (not negative)
- \( r^3 < 0 \) (negative)
- \( r^4 > 0 \) (not negative)
- \( r^2 \times r^3 < 0 \) (negative)

Thus, the expressions that must be negative are \( r^3 \) and \( r^2 \times r^3 \).

### Final Answer:
The expressions that must also be negative are:
- \( r^3 \)
- \( r^2 \times r^3 \)