Whether it's removing live roots that are causing structural damage to concrete and piping, or removing the roots from a felled tree, there are a handful of tools to aid in uprooting the roots
Depending on how large and numerous the roots are, a combination of hand tools can be used. But be prepared to work up a sweat; this is no easy task. They include a landscape bar (sometimes called an all-purpose utility bar), heavy duty digging spades and mattocks (similar to a pickaxe). Save your hands and pick up a good pair of work gloves, as well.
Renting a machine will save your back and your sanity. For larger trees, a small backhoe or skid steer may be necessary, but if yard space doesn't permit, a stump grinder should do the trick. Chainsaws can be used, but are not recommended. Always wear protective eyewear when operating machinery.
If you've got some time, there are chemicals available that hasten the rotting process. Drill several holes in the roots or stump and fill with the chemical. This process can take a long time to produce results.
Only legally an option when seasonal conditions permit, start a fire on top of where the roots meet the stump. Be prepared to make a weekend of stoking the fire, because a stump can smoulder for days.
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Keep in mind that there will be some unforeseen consequences to lopping of roots to a live tree. Because it is impossible to tell how the tree will react, you should see if there is any way to move forward without removing live roots.