(* Example 1-- nx=1.2, ny = 1.3, nz = 1.4, what is n in the (1,1,1) direction? *)
Solve[1/n^2 == (1/3)/(n^2-1.2^2) + (1/3)/(n^2-1.3^2) +(1/3)/(n^2-1.4^2) ,n]

Plot[{1/n^2 , (1/3)/(n^2-1.2^2) + (1/3)/(n^2-1.3^2) +(1/3)/(n^2-1.4^2)},{n,1.1,1.5}, PlotRange-> {-10,10}]

Plot[{1/n^2 , (1/3)/(n^2-1.2^2) + (1/3)/(n^2-1.3^2) +(1/3)/(n^2-1.4^2)},{n,1.1,1.5}, PlotRange-> {0,2}, Exclusions->{n==1.2,n==1.3,n==1.4}]

FindRoot[1/n^2 == (1/3)/(n^2-1.2^2) + (1/3)/(n^2-1.3^2) +(1/3)/(n^2-1.4^2) ,{n,1.24}]
FindRoot[1/n^2 == (1/3)/(n^2-1.2^2) + (1/3)/(n^2-1.3^2) +(1/3)/(n^2-1.4^2) ,{n,1.35}]