Pentecontaetia is a really big fancy word that describes a time a long, long time ago in ancient Greece. Think of it like a really long story that happened over 50 years!
Back then, there were two big cities called Athens and Sparta, and they didn't always get along. They were both very powerful and wanted to be the best, but sometimes they would fight with each other.
During the Pentecontaetia, the city of Athens became very strong and important. They built a big navy with lots of boats to protect themselves from Sparta and other enemies. They also started to make other cities around them join with them, kind of like when you make new friends at school.
But Sparta didn't like this. They thought Athens was getting too powerful and were afraid they might try to take over other cities. So Sparta and Athens fought a big war called the Peloponnesian War, which lasted for a long time and was very hard for both sides.
In the end, Sparta won the war and Athens became less powerful. But the Pentecontaetia was still an important time in history because it showed how two powerful cities could try to outdo each other and sometimes things don't always turn out the way we wanted them to.