x^(1/3) ... is it the same as x^(2/6) ?

lets consider x < 0

x^(1/3) is okay

(-8)^(1/3) = -2

then

x^(1/3) = (x^2)^(1/6) is okay since (-8)^2 = 64^(1/6) ((but this = 2))

and finally

x^(1/3) = (x^(1/6)^2) is not okay since (-8)^(1/6) = undefined

does anyone have an explanation, idea or comment about this? my math teacher brought this up to me today