Actually, yes! Air plants do grow roots, although they do not serve the same purpose as the typical plant root.
Air plants use their roots for attaching themselves to rocks, trees, shrubs, or wherever they live. Unlike other plants, they do not need roots to take in nutrients or water – they do that all through their specialized leaves.
If you don’t like the look of the roots, you can cut them off without harming the plant. In fact, many retailers do just that.
However, I usually leave my roots alone. Sometimes the roots provide helpful ways of attaching or mounting the plant to its decorative home. Also, if you are lucky enough that your air plants start to grow into clumps, the roots help keep all of them together.
If your air plant starts to grow roots from its base, take that as a compliment that it is happy where you have placed it and wants to stay there!