Question
I am replacing my sidewalk on the side of my house and the roots from several trees from my neighbors yard are just below the surface that caused the uplifting in the sidewalk.  My neighbor said it was okay to cut them out, but one major root is about a foot below the surface going under my slab foundation to the house.  The tree is approximately 3 feet from the fence and the fence is 5 feet from our house.  We have a lot of hardpan (compact clay soil) and the roots have a difficult time growing down.  My construction crew plans on pouring cement for the new sidewalk next Tuesday, but they didn't want to cut the big roots thinking it would have a negative result on the tree.  How damaging would it be to the trees to cut the big roots?

Answer
The roots of a tree will extend about 1 1/2 times the width of the branches. A tree can survive root cutting if the amount of roots cut does not exceed 25-30% of the root system. This includes adding soil over the root area or taking soil away from the root area. The root system can be measure roughly by drawing a circle around the tree where the ends of the branches hang and then estimate the area to be cut or filled and calculate the percent damage. The roots will extend further than the branches but these will be smaller and can with stand cutting. The other thing IF you do this MAKE SURE you rope off the rest of the tree and the roots zone so NO traffic of machinery goes in this zone --this will protect what is left of the root system from compaction etc. IF you cut a large roots you can expect some foliage dieback on that side of the tree. You have cut the root that supplies the food and water to the foliage and the tree will drop some of the foliage to compensate for the root loss.
This will not be a serious health issue for the tree but if left it could cause the more damage to the sidewalk later. I would cut he roots.