The correct answer to this open question is the following.
Although there are no options attached we can say the following.
By what process did the Muslim Empire receive and exchange their ideas, achievements, and knowledge?
We can say that the Muslim Empire received and exchanged those things through trade and conquest. But basically through trade.
Islam spread in the 13th century.
In the beginning, Islam was introduced through Arab Muslim traders and merchants. After they did so, it was the rulers through the conversion of the elite people. Missionaries came from Southern Asia, like Champa and Gujarat. There also were missionaries from the Hadhramaunt, in Arabian Peninsula.
In those years, during the Fatimid Caliphate, Muslims from Arabi expanded their territories and captured some parts of North Africa, including Egypt. Trade was a very important activity and helped spread Islam throughout many regions of Africa. Muslim traders took their Islam beliefs everywhere and influenced many people.