The tubers are actually the swollen ends of rhizomes, another modified, underground stem structure. To find the roots you have to follow the stem down from above ground and carefully dig up the entire plant. Somewhere in the center of the cluster of rhizomes with their tubers you will find a taproot. In fact, anyone who leaves potatoes around too long discovers that the eyes turn into sprouts eventually. And to plant potatoes all you have to do is cut up a potato, making sure each fragment has eye, and then bury them.
Many plants have rhizomes without tubers.
Irises are a well known example. Dig up an iris and you discover that it has very thick rhizomes that are used to reproduce. As the rhizome grows laterally underground it eventually produces a new bud which grows upward and produces a new group of leaves and flowers. Gardeners often take advantage of this and split the rhizomes manually to propagate irises and other rhizomatous perennials. This same tendency can be a detriment to gardeners.