Answer: First, try to determine whether the root is from a nearby tree or shrub. If it is, you won't want to hurt the parent plant by using a vegetation killer without first severing the root from the parent plant. Next, walk to the other side of the cement strip and dig straight down, far enough to cut through the root. The part that travels beneath the cement and comes up on the other side through the crack can then be treated with a product like Round-Up. Brush the vegetation killer directly on the root that's coming up through the crack. This should kill it completely.
Best wishes with your landscape!
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