M23-05

Official Solution:


If \(a\) and \(b\) are integers and \(ab \neq 0\), is \(a^b\) positive?

First, note that \(ab \neq 0\) implies that neither variable is 0.

(1) \(b^a\) is positive.

If \(b\) is positive, \(a\) can be either positive or negative. In this case, we can obtain both YES and NO answers to the question. For instance, if \(b = a = 1\), then \(a^b\) will be positive. However, if \(b = 1\) and \(a = -1\), then \(a^b\) will be negative. It is also worth noting that if \(b\) is a positive even number, \(a^b\) will be positive regardless of the value of \(a\). For example, if \(b = 2\), then \(a^b\) will be positive, no matter the value of \(a\).

Since we have established that \(a^b\) can be both positive and negative when \(b\) is positive, there is no need to further investigate the first statement. However, for the sake of learning, let's delve into the scenario where \(b\) is negative. If \(b\) is negative, for \(b^a\) to be positive, \(a\) must be an even number, whether positive or negative. In this case, we can also obtain both YES and NO answers. For example, if \(b = -1\), and \(a = 2\), then \(a^b\) will be positive. However, if \(b = -1\), and \(a = -2\), then \(a^b\) will be negative. Similarly, it is worth noting that if \(b\) is a negative even number, \(a^b\) will be positive regardless of the value of an even \(a\). For example, if \(b = -2\), then \(a^b\) will be positive, no matter the value of an even \(a\).

Not sufficient.

(2) \(b\) is negative. This statement is clearly insufficient.

(1)+(2) As discussed above, when \(b\) is negative, we can obtain both YES and NO answers to the question. For example, if \(b = -1\), and \(a = 2\), then \(a^b\) will be positive. However, if \(b = -1\), and \(a = -2\), then \(a^b\) will be negative. Not sufficient.


Answer: E