Question

What is the sign of
\[m^{13} \cdot n^{5}\] ?
A number line with 4 evenly spaced tick marks. The second tick mark is labeled 0. A closed point labeled m is plotted to the left of first tick mark. A closed point labeled n is plotted between the first tick mark and 0.
\[0\]
\[m\]
\[n\]
Choose 1 answer:
Choose 1 answer:
(Choice A) Positive
A
Positive
(Choice B) Negative
B
Negative
(Choice C) Zero
C
Zero

Answer 1

To determine the sign of \(m^{13} \cdot n^{5}\), we need to analyze the signs of \(m\) and \(n\) based on their described positions on the number line.

1. **Position of \(m\)**: It is plotted to the left of the first tick mark, which means \(m\) is negative (since it is less than 0).

2. **Position of \(n\)**: It is between the first tick mark and 0, which implies that \(n\) is also negative (since it is greater than 0 but still less than the first tick mark).

Now we compute the signs for \(m^{13}\) and \(n^{5}\):

- **Sign of \(m^{13}\)**: Since \(m\) is negative and raised to an odd power (13), \(m^{13}\) will be negative.

- **Sign of \(n^{5}\)**: Since \(n\) is negative and raised to an odd power (5), \(n^{5}\) will also be negative.

Now, we multiply the two results:

\[
m^{13} \cdot n^{5} = (\text{negative}) \cdot (\text{negative}) = \text{positive}
\]

Thus, the sign of \(m^{13} \cdot n^{5}\) is positive.

The correct answer is:
**(Choice A) Positive**