Although labor-intensive, this directly addresses the issue with no risk to nearby plants. Dig out the soil around and under the root first, then cut with a root saw or loppers. Cutting roots aggressively can put the tree in a long decline, potentially killing it over many years. Follow these rules of thumb to prevent this:  Multiply the diameter of the tree by eight. That's the minimum distance from the trunk you can cut with low risk of serious harm.  Cut roots on one side of the tree only, especially if you're risking a cut closer than the minimum distance.
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