For　a　given　undirected　(edge)　weighted　graph　G　=　(V,　E),　a　terminal　set　S　subset　of　V　and　a　root　r　is　an　element　of　S,　the　rooted　k-vertex　connected　minimum　Steiner　network　(kVSMN(r))　problem　requires　to　construct　a　minimum-cost　subgraph　of　G　such　that　each　terminal　in　S　\　{r}　is　k-vertex　connected　to　r.　As　an　important　problem　in　survivable　network　design,　the　kVSMNr　problem　is　known　to　be　NP-hard　even　when　k　=　1　[14].　For　k　=　3　this　paper　presents　a　simple　combinatorial　eight-approximation　algorithm,　improving　the　known　best　ratio　14　of　Nutov　[20].　Our　algorithm　constructs　an　approximate　3VSMNr　through　augmenting　a　two-vertex　connected　counterpart　with　additional　edges　of　bounded　cost　to　the　optimal.　We　prove　that　the　total　cost　of　the　added　edges　is　at　most　six　times　of　the　optimal　by　showing　that　the　edges　in　a　3VSMNr　compose　a　subgraph　containing　our　solution　in　such　a　way　that　each　edge　appears　in　the　subgraph　at　most　six　times.