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.
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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).
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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 \(n^{5}\):
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Sign of \(m^{13}\): Since \(m\) is negative and raised to an odd power (13), \(m^{13}\) will be negative.
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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
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