You have six such solutions, each a parametrized branch of polynomial roots.
In Mathematica:
solns = ff /. Solve[dd == ff/kk + z*(ff/rr)^6, ff]
(* Out[73]= {Root[dd - #1/kk - (z #1^6)/rr^6 &, 1],
Root[dd - #1/kk - (z #1^6)/rr^6 &, 2],
Root[dd - #1/kk - (z #1^6)/rr^6 &, 3],
Root[dd - #1/kk - (z #1^6)/rr^6 &, 4],
Root[dd - #1/kk - (z #1^6)/rr^6 &, 5],
Root[dd - #1/kk - (z #1^6)/rr^6 &, 6]} *)
One can give the parameters actual numeric values in order to get numeric solutions. For example:
solns /. {dd -> 3.3, kk -> 5.2, rr -> 1.07, z -> 2.8}
(* Out[74]= {-1.11127, 1.08777, -0.544166 - 0.962542 I, -0.544166 +
0.962542 I, 0.555916 - 0.942204 I, 0.555916 + 0.942204 I} *)
