Pull up the finished flooring near where you intend to add concrete to the slab floor. Pull up interlocking wooden slats with a pry bar, exposing the plywood subfloor beneath. The subfloor is screwed into the sleeper joists beneath it. Remove the screws with the drill, setting them to one side.
Remove the subfloor to reveal the sleeper joists below. Sleeper joists are typically pressure-treated 2-by-4s that rest on a plastic moisture-barrier. They are screwed into the concrete slab below with masonry screws. Remove the screws with the drill and the masonry bit, then remove the sleeper joists to clear the area where you will add concrete.
Cut out portions of long floor joists that run the length of the room instead of pulling up the whole floor to get rid of the entire 2-by-4. Pull up as much of the subfloor as necessary, exposing the joists. Cut through the joists with a handsaw at the point where they need to be removed. Unscrew those portions of the joists and remove them.
Cut through the plastic moisture barrier with the utility knife. Only cut away the portion that must be removed; leave the rest intact.