As lgdry15 got the correct answer I will perform a standard trick towards the solution.
Let u = (k-x)/x, v = (k-y)/y, w = (k-z)/z (The case for x/y/z = 0 is obvious, so WLOG suppose they are positive.)
Then x = k/(u+1), y = k/(v+1), z = k/(w+1)
So that the equality becomes uvw = 1 and to show that (u+1)(v+1)(w+1) >= 8k^3.
Since u,v,w > 0, the result is obvious by AM-GM.
Let u = (k-x)/x, v = (k-y)/y, w = (k-z)/z (The case for x/y/z = 0 is obvious, so WLOG suppose they are positive.)
Then x = k/(u+1), y = k/(v+1), z = k/(w+1)
So that the equality becomes uvw = 1 and to show that (u+1)(v+1)(w+1) >= 8k^3.
Since u,v,w > 0, the result is obvious by AM-GM.