M30-09

Official Solution:


What is the value of nonzero integer \(k\)?

(1) \(|k| + k = 0\).

Rearranging: \(|k|=-k\). This implies that \(k\leq{0}\). Since we are told that \(k\) is a nonzero integer, then we have that \(k < 0\). However, this is not sufficient to determine the value of \(k\).

(2) \(|k^k| = k^0\).

Any nonzero number raised to the power of 0 is 1, so \(k^0=1\). Thus, we have \(|k^k| = 1\). This implies that \(k=1\) or \(k=-1\). Again, this information is not sufficient to determine the value of \(k\).

(1)+(2) Since from (1) \(k < 0\), then from (2) \(k=-1\). Sufficient.

(1) + (2) Since from (1) \(k < 0\), and from (2) we have the possibilities \(k=1\) or \(k=-1\), we can deduce that \(k=-1\). Sufficient.


Answer: C