For a connected graph and a positive integer , the th power of is the graph with where if the distance between and is at most . The edge coloring of defined by assigning each edge of the color produces an edge-colored graph called a distance-colored graph. A distance-colored graph is properly -connected if every two distinct vertices and in the graph are connected by internally disjoint properly colored - paths. It is shown that is properly -connected for every -connected graph that is not complete, a double star is the only tree for which is properly -connected, and is properly -connected for every connected graph of diameter at least . All pairs of positive integers for which is properly -connected are determined. It is shown that every properly colored graph with colors is a subgraph of some distance-colored graph and the question of determining the smallest order of such a graph is studied.