Although you've got to be an Apple Music subscriber to watch the entire episode, the first 6 minutes (above) were shared on Corden's YouTube page.
The clip starts with the duo riding around Los Angeles jamming to Smith's "Getting Jiggy With It," then they talked about the actor's first single, "Girls Ain't Nothing But Trouble," which hit the radio when Smith was just a senior in high school.
Corden also asked Smith if he's ever been offered the role of Barack Obama, and the movie star -- who has played real-life figures including Muhammad Ali, investor Chris Gardner and neuropathologist Dr. Bennet Omalu -- said he actually spoke to the former POTUS about the possibility.
"He told me felt confident that I had 'the ears' to play the role," Smith said.
You can watch "Carpool Karaoke" exclusive at applemusic.com/carpoolkaraoke