Final answer:
Ancient city-states and empires primarily used diplomacy, including alliances and intermarriage, as well as trade, to peacefully acquire territory and expand their influence.
Step-by-step explanation:
The ancient city-states and empires primarily used diplomacy to acquire territory through peaceful means. This included establishing alliances with other states, which could involve actions such as intermarriage between royal families, or the paying of tribute to a more powerful state. For example, trade was a major factor that brought about peaceful relations and territorial expansion, as seen in the unifying of the African continent by means of the Arab invasion and Bantu migration, facilitated by the brisk trade in gold salt and slaves. In all cases diplomacy and trade were the preferred methods for peaceful territorial expansion.