It matters what you mean by "break down". A tap root can split a rock based on the rock's composition and facilitate its change. Since matter is never lost, the rock will only deplete into smaller chucks. A good example of this is in Southern Utah. The giant mesas look like their solid rock (and the are definitely hard to the touch), but they have a sparse amount of plants and trees that crop out of them. These plants are rugged desert plants that live off of the water that seeps into the rock. Of course, the water takes 14,000 years to get through the rock, but there is enough retained that a root system can be formed, in turn affecting the integrity of the rock.
