Final answer:
The compass, astrolabe, and caravel were the technologies that allowed the Portuguese to set up their trade empire in the Indian Ocean region.
Step-by-step explanation:
The piece of technology that allowed the Portuguese to set up their trade empire throughout the Indian Ocean region was the compass, the astrolabe, and the caravel.
These technological advancements enabled the Portuguese to navigate the open sea and explore new trade routes.
Under the sponsorship of Prince Henry the Navigator, the Portuguese established trading posts in Africa and later expanded to India and the Spice Islands, marking the beginning of their powerful sea empire.