Final answer:
Tiglath-Pileser III was the Assyrian king who extended the empire and made Israel a vassal state between 745-727 BC.
Step-by-step explanation:
The Assyrian king from 745-727 BC who extended the empire and made Israel a vassal state was Tiglath-Pileser III. During this time, the Neo-Assyrian Empire grew in power and dominance across the Near East. Under Tiglath-Pileser III's leadership, the empire exerted control over territories in Mesopotamia and the eastern Mediterranean, including the conquest of the Aramaean kingdoms in Syria and Babylonia, before eventually establishing dominion over Israel.