How did the persians build a strong empire

Per the question in your picture:

C-Athens had defeated Persia at the Battle of Salamis, so Persia supported Sparta.

Explanation:

The Peloponnesian Wars were between Athens and Sparta (Greek).  Persia supported Sparta because Athens had defeated Persian so they became allies with Sparta.

How did the persians build a strong empire?  By using a common currency

i think your answer there is c