Create Root Cuttings that can be used to grow another plant
The main central taproot of the horseradish should be left as unmolested as feasibly possible. You can either replant it, store it, or grind it up for a culinary endeavor. The thinner side roots coming off the taproot should be cut into 6 to 8 inch long sections that are at least 1/4 inch in diameter.
Planting the Root Cuttings
Plant the root cutting upside down at an angle, with the top of the cutting about 2 inches below the ground. Spacing should be 2 to 2.5 feet apart in all directions.
Put some water in the hole with the cutting before covering it over. Once your cuttings are covered water them again thoroughly, be sure the water saturates the ground to a depth of about 3 or 4 inches. You're not making a duck pond but you want the ground wet wet wet for now at least.
Once watered place a healthy layer of organic mulch such as bark, wood chips, straw in the space between the plantings but leave a separation of an inch or two between the mulch and where the cuttings are planted.
Create Crown cuttings or Divisions to Replant
Crown cuttings are simpler than root cuttings. For crown cuttings, cut the plant into equal sections. All portions should have a fair share of both foliage and roots.
Replanting the crown is pretty much the same as planting a transplant, be sure to water them thoroughly as described under root cuttings [above] and mulch them well also.