How can I trim roots?

Surface roots are always a problem. Depending on how much you want to remove or cut, your tree will be fine/or not. If it is just one root that is breaking the sidewalk, you should be fine. More than 20% of the root system, would cause trouble for the tree. Remember that roots spread, especially surface roots - so cutting what is causing the problem as far back from the sidewalk as possible and removing that section will work. (Think of the surface of the ground like a mirror and you can get a good idea of your tree's root system under the soil.) You can also get a steel plate and sink it a few feet away from your new concrete (as long as you're digging) to prevent similar issues in the future.

Great question! But a tough one. This could be happening for any number of reasons. Bad placement, improper pruning, drought, etc. On many trees if you trim the surface roots, you risk the tree dying or falling over in high winds. See if these help.

https://homeguides.sfgate.com/sidewalks-tree-growth-52028.html

https://inexpensivetreecare.com/blog/what-to-do-when-your-trees-roots-are-destroying-the-sidewalk/

If it were at my house, I would have a consult with a licensed arborist to determine how to proceed. When I bought my current home may years ago, the prior owners placed several trees next to the property line, never considering mature growth. I had to take them all down as they were going to lift up stone walls that it would have been my responsibility to repair/replace which I couldn't afford to do. It broke my heart, but had to be done.

In your case, if you trim the roots and replace the concrete, you run the risk of having to do it again in several years as the tree continues to grow.

Hyrik...My dad used an axe and just chopped them out because they were hard to mow over. It was a silver maple and got so bad that he finally cut it down.

No. You cannot remove the Roots. Instead you replace the Concrete with Bricks.