Establishment of the Satavahana Empire:
The Satavahana Empire was established around the 2nd century BCE in the Deccan region of present-day India. The exact origins of the Satavahanas are unclear, but they are believed to have been a local dynasty that rose to prominence in the wake of the Mauryan Empire's decline. The founder of the empire is thought to be Simuka, although some sources suggest that there were earlier rulers.

Political Expansion:
Under the rule of Simuka and his successors, the Satavahana Empire witnessed significant political expansion. They adopted a policy of expansion through military conquests as well as strategic alliances. The empire was known for its efficient administration and a well-organized bureaucracy.

Extent of the Empire:
The Satavahana Empire reached its zenith during the reign of Gautamiputra Satakarni, who is considered one of the greatest Satavahana rulers. The empire stretched from the Godavari River in the north to the Krishna River in the south, and from the Arabian Sea in the west to the Bay of Bengal in the east. It encompassed large parts of present-day Maharashtra, Andhra Pradesh, Telangana, Karnataka, and Madhya Pradesh.

Administration:
The Satavahana Empire had a decentralized administrative system. The empire was divided into several provinces called Ahara or Bhoga, each headed by a governor. The governors were responsible for collecting taxes, maintaining law and order, and administering justice. The empire also had a well-established system of taxation, with revenue collected from land, trade, and other sources.

Trade and Economy:
The Satavahanas played a crucial role in facilitating trade between North India and South India. They controlled major trade routes that connected the Deccan region with the western coast and the eastern coast. The empire was known for its flourishing maritime trade, with Roman coins and other artifacts being found in excavations. The Satavahanas also minted their own coins, which were widely circulated.

Cultural and Religious Influence:
The Satavahana Empire was a melting pot of various cultures and religions. They patronized Buddhism, Jainism, and Hinduism, and their rule witnessed the coexistence of these religions. The empire also witnessed the development of art and architecture, with cave temples and stupas being constructed during this period.

In conclusion, the Satavahana Empire was established in the 2nd century BCE and expanded its political control over a vast territory in the Deccan region of India. The empire had a decentralized administrative system, promoted trade and commerce, and patronized various religious and cultural traditions. Its legacy can still be seen in the art, architecture, and cultural practices of the region.