A root is the part of the plant or tree in the soil.
“The roots of this trees have grown under the sidewalk.”
It can also be used as an adjective for the category or vegetables like carrots, beets, and parsnips: these are root vegetables.
Rout means a disorderly retreat or a disastrous defeat.
“Arsenal didn’t stand a chance against Manchester United. The final score of 6-1 showed that the game was a rout.”
A route is the path you take to get somewhere.
“Google maps shows three different routes from Chicago to Los Angeles.”