We study transmit power minimization in the two-user Z Interference Channel (ZIC). When the interference link gain is strong, the capacity of the ZIC has been fully characterized. For this strong interference regime, we derive the closed-form solution of the minimum required transmit power to achieve a given rate pair, and show that the resulting power allocation between the two users is not necessarily unique. When the interference link gain is weak, the capacity of the ZIC is still an open problem to date. For this weak interference regime, we develop an inner and outer bound for the required transmit power to achieve a given rate pair, and characterize the constant power ratio relation between the two bounds. In contrast to the strong interference case, rate splitting is necessary for power minimization in this regime. The optimal power-minimizing rate splitting solution is then derived, and performance in terms of total transmit power required for a given rate pair is analyzed.