What happened to the jewish people when they were living in egypt?

According to Jewish tradition, the Israelites were in Egypt for 210 years (Rashi commentary, on Exodus ch.12). For the first century, they lived in peace and plenty, as long as Joseph and his brothers were still living. After that, they were enslaved by the Egyptians. The enslavement was gradual, in steps, eventually becoming harsh and crushing (Exodus ch.1). At the end of that time, God sent Moses (Exodus ch.3) to demand that Pharaoh free the Israelites; and He brought plagues upon Egypt (Exodus ch.7-12) when Pharaoh refused.  

As a related topic, a brief biography of Moses may be pertinent here:

Moses was born 245 years after the death of Abraham. Pharaoh had decreed that Israelite boys be killed (Exodus ch.1), but the daughter of Pharaoh took pity on the infant Moses (Exodus ch.2) and raised him as her own son. He was forced to flee after killing a cruel Egyptian taskmaster, and went to Midian, where he wedded the daughter of Jethro. He eventually achieved the highest level of prophecy (Deuteronomy ch.34) and was called upon by God (Exodus ch.3).  

The importance of Moses cannot be overstated. He brought the Israelites out of Egyptian slavery (Exodus ch.12). He received the Torah from God (Exodus 24:12) and later recorded it in writing (Deuteronomy 31:24). He went up on Mount Sinai for 40 days and nights (Deuteronomy ch.9-10) and brought down the Two Stone Tablets with the Ten Commandments (Exodus 31:18). He brought the Israelites into the covenant with God (Exodus ch.19 and ch.24), and he oversaw the building of the Tabernacle (Exodus ch.35-40). He was the humblest of men and the greatest of prophets (Numbers ch.12).

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