x5/3-6x-x4/3=0.
Let y3=x, then y5-6y3-y4=0=y3(y2-y-6)=y3(y-3)(y+2), so y=0, 3 or -2.
Therefore x=0, 27, -8.
CHECK
x=0 satisfies the original equation.
When x=27, x5/3=35=243; x4/3=34=81; 6x=162.
x5/3-6x=243-162=81=x4/3.
When x=-8, x5/3=(-2)5=-32; x4/3=(-2)4=16; 6x=-48.
x5/3-6x=-32+48=16=x4/3.
So solution is confirmed.