8¹/³ : (-1¹/⁶)

To simplify the expression 8^(1/3) : (-1)^(1/6), we can first rewrite the numbers as base numbers raised to the given fractions.

8^(1/3) can be written as the cube root of 8, which is 2, since 2^3 = 8.

Similarly, (-1)^(1/6) is the sixth root of -1, which has two solutions considering its properties: (-1)^(1/6) = exp(iπ/6) = cos(π/6) + i sin(π/6) = (sqrt(3))/2 + i/2 or (-1)^(1/6) = exp(5iπ/6) = cos(5π/6) + i sin(5π/6) = - (sqrt(3))/2 + i/2.

So the expression simplifies to: 8^(1/3) : (-1)^(1/6) = 2 : ((sqrt(3))/2 + i/2) or 2 : (-(sqrt(3))/2 + i/2)

Please note that the relationship between two complex numbers and their value depends on the perspective or context in which they are considered.