To take root; to become established. Used literally for plants and figuratively for customs, ideas, or systems that become firmly embedded in a community or society.
その風習は長い年月をかけて地域に根付いた。
That custom has taken root in the region over many years.
新しい技術が職場に根付くには時間が必要だ。
It takes time for new technology to become established in the workplace.
若者のあいだでこの言葉がすぐに根付いた。
This word quickly caught on and became established among young people.