Final answer:
The Jews were allowed to return to their homeland after the Babylonian exile because Babylonia was conquered by Persia and the Persian king Cyrus the Great permitted their return.
Step-by-step explanation:
The Jews were allowed to return to their homeland after they were exiled to Babylon because Babylonia was conquered by Persia. The Persian king Cyrus the Great permitted the unassimilated Judeans to return to Judah and they began the process of reconstruction, including the rebuilding of the temple in Jerusalem.