MikeRob--If you are talking about I-40, it is closed and has been for many weeks. I'm not sure of the current situation but I think it will be well into March, at best, before it opens. Your only real alternative from the Tennessee side is I-81 North out of Knoxville to I-26, then into Asheville to connect with I-40, then I-40 West to Hwy. 74 towards Waynesville and Bryson City. There's a Cherokee exit six or eight miles before you get to Bryson City.
You could go down through Maryville and come back northeastwards on the road through Deals Gap ("Dragon's Tail") but I wouldn't recommend it. Hwy. 441 has been off and on for some time, as is typical in winter. In other words, it's about as close to a real-life case of "you can't get there from here" as you are likely to encounter.