Congratulations on your new home!* You might want to use a pruning saw to cut the roots as close to the concrete as you can and then pull them out.* Cutting the roots in sections might work, even if you can't cut all the way through.* That will help them to rot or make it easier to pull out.*
Another option would be to see if you can rent one of these.
If you find you need to purchase one and can't rent it, and won't have any further need for it, consider donating it to a local native plant society or garden group.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.