Official Solution: The question cannot be easily rephrased to incorporate the particular information given. However, of course we should take note that both variables are integers and that \(x\) is less than \(y\). We are looking for the value of \(x + y\).
Statement (1): SUFFICIENT. First, we should list out all the possible scenarios in which integers \(x\) and \(y\) fit the equation \(x^y = 4\).
There are three possibilities, as we can find by trial and error: \(2^2 = 4\), \((-2)^2 = 4\), and \(4^1 = 4\). However, of these possibilities, there is only one for which \(x\) is less than \(y\), namely \((-2)^2 = 4\). Thus, we can find the value of \(x + y\), which is \(-2 + 2 = 0\).
Statement (2): SUFFICIENT. Knowing that \(|x| = |y|\) does not tell us the values of the integers. However, since they have the same absolute value, but \(x\) is less than \(y\), it must be the case that \(y\) is a positive integer and \(x\) is the negative of that integer. For instance, if \(y\) is 5, then \(x\) is -5. The sum of \(x\) and \(y\) must therefore be 0, no matter what.
Answer: D
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