It is common to see a batter drop his bat in fair territory in the area in front of home plate. Usually this results in no further complication of the play. But when the dropped bat touches a batted ball, every umpire, coach and player should know the rule

Ball rolls into dropped bat (foul)

If the bat is dropped in foul territory and the ball rolls into the bat and then goes into fair territory, the ball is ruled foul. Upon touching the bat in foul territory, the ball is foul and is a dead ball.

Batter drops bat in foul territory but the ball touches bat and rolls into fair territory