Place the pork roast in the slow cooker. Stir together 1/2 cup of BBQ sauce and 2.5 cups of root beer. Pour the root beer mix over the pork roast in the slow cooker. Sprinkle the garlic powder and smoked paprika over top.
Secure the lid on the slow cooker and cook on low for eight hours.
After eight hours, use tongs and a slotted spoon to remove the meat and bone from the liquid in the slow cooker (the bones can be discarded). The meat will be extremely tender and should easily separate from the bone.
Shred the meat using a fork and either serve as is or mix in the remaining BBQ sauce.