It sounds like what you are calling a root might be the stolon, that attaches it to the hen/rosette.
Another thing about sempervivum, they do not do well growing indoors. They are an alpine type plant and really need the cold changes of winter in order to thrive. With you living in Zone 8 it seems that you would have some winter type weather? Perfect.
Can you take the offset out of the pot and take a picture of the underside, so I can see if it is starting to form new roots?
You also might have better results with using a sandy loam mix to plant them in. You can get a small bag of Miracle Grow planting mix, add some coarse sand and lay the offsets on top of the new mix. Make sure you don't get the one that has the moisture holding beads in it. What I see in your container shows a mix that is mainly made from tree products and may some moss? If you think about where our sempervivum originate from you realize that they prefer rock type products, like sand, rock dust, grit/gravel.
You can start lightly misting them once you see new white roots forming.
Jasmine, sending you treemail.
