Answer: I'm afraid there is no easy solution. As you've discovered, the roots can go quite deep and the only way to get rid of them is to dig them up. Or, as they appear you can cut them down to ground level and then immediately paint full strength RoundUp on the cut ends. This should effectively kill that particular root. Keeping the sprouts chopped down - before they have a chance to develop leaves - will stop photosynthesis and effectively starve the roots out. It just takes persistence. Wish there were an easier way!
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