What was the outcome of the third punic

war and how did they know there would be no fourth?

Rome defeated Carthage and gained control over the Mediterranean.

Explanation:

Between the period of 149 and 146 BCE, the Third Punic War was fought between Carthage and Rome. Carthage already had defeated in two previous wars against Rome. Lasting just 3 years or so, it witnessed the utter collapse of the Carthaginian and Punic empires as a whole It led to the assimilation of North Africa and the rest of the Punic territories as Roman.

They knew that there would be no further war because Carthage was crushed, its people were sold into slavery. Carthage was removed as a potential threat to Rome, and Rome gains control over the entire Mediterranean region and extends its empire.