Final answer:
The nurse should delegate the assistive personnel to transport a client to the radiology department for an x-ray first, as it is time-sensitive and within the AP's scope of practice.
Step-by-step explanation:
When considering which tasks a nurse should delegate to an assistive personnel (AP), it is essential to prioritize based on the urgency and scope of practice. Among the tasks listed, the nurse should delegate the AP to transport a client to the radiology department for an x-ray first. This task requires timely coordination for diagnostic procedures and involves client safety and movement, which are within the scope of the AP's responsibilities. Other tasks such as passing fresh water to clients and obtaining a routine urine sample, while important, are less time-sensitive. The task of taking an arterial blood gas (ABG) specimen to the laboratory is also important, but does not require immediate action like a scheduled radiology appointment might.