Final answer:
The Ptolemaic rulers legitimized their rule by adopting Egyptian religious practices, fulfilling the role of pharaohs, and blending Greek and Egyptian cultures, particularly through the creation of the deity Serapis.
Step-by-step explanation:
The Ptolemaic rulers sought to legitimate their rule in Egypt by adopting and respecting the religious practices and cultural customs of the native Egyptians. They positioned themselves as the successors to Alexander the Great, embracing the role of pharaohs, and integrating Greek culture, which remained as the language of the elites, while also using Egyptian officials in the governance of their state.
The Ptolemies even went so far as to create a new deity, Serapis, which blended Greek and Egyptian religious traditions, thereby allowing the disparate subjects of Ptolemaic Egypt to find common ground in worship. The correct answer to the question concerning how the Ptolemaic rulers of Egypt sought to legitimate their rule is: b. by adopting Egyptian religious practice and styling themselves as pharaohs.