Final answer:
Cyrus the Persian allowed the exiles to return to Judah and rebuild the temple in Jerusalem in the year 539 BCE.
Step-by-step explanation:
The year when Cyrus the Persian, the conqueror of Babylon, permitted the exiles to return to Judah and rebuild the temple in Jerusalem at imperial expense was in 539 BCE. Following his conquest of the Neo-Babylonian Empire, Cyrus the Great allowed all the enslaved individuals of the Babylonians, including the Jews, to return to their homelands. This marked the end of the Babylonian Captivity and began the return to Judah, where the Jews rebuilt the Temple. Additionally, many Jews chose to remain in Babylon and, soon, other cities in the Persian Empire, leading to the beginning of the Jewish diaspora.