The zeroth-order general Randic index,0Rα(G), of a connected graph G, is defined as (formula presented), where di is the degree of the vertex vi of G and a arbitrary real number. We consider linear combinations of the0Rα(G) of the form0Rα(G) - (Δ + δ)0Rα-1(G) + Δδ0Rα-2(G) and0Rα(G) - 2a0Rα-1(G) + α2 0Rα-2(G), where a is an arbitrary real number, and determine their bounds. As corollaries, various upper and lower bounds of0Rα(G) and indices that represent some special cases of °Ra(G) are obtained.