The Medici family didn't start so powerful they gained there power. The Medici family was a family of bankers and lived in Florence, Italy in the 14 century. They were like any other family but they gained more power by bribing people, causing violence, and killing people that were in there way. They supported many of the great artists and thinkers like Brunelleschi, Leonardo da Vinci, Michelangelo, and Galileo. They were very interested in the arts and secret ideas from the Romans.
The family got even popular when the Pope said that the Medici's bank was the most reliable bank. They earned so much money that some of the other banks went out of business. They supported Brunelleschi but most people didn't support him. Although, when he started and finished building the dome people were amazed of his work.
I think the Medici family had a great impact because if it wasn't for the Medici family the dome they were trying to build wouldn't even be finished. Also they brought back classical arts.