@Alycat727 - If you want to do 3 days on trail you could start on either end of the trail in the park.. Fontana Dam or Davenport Gap and hike to the dome. Your pace would determine how many miles you'd need to hike each day to get to the dome from either end. Obviously if you're doing 3 days, then no... you can't slackpack because you'll need your gear.
If you just want to see the dome without packs then just drive up to it and walk the 0.5 mile paved Pathway up to it. Easy peasy.