What is the value of n?

Do you mean (1/3)⁸ = (-1/3)ⁿ - 1 or  (1/3)⁸ = (-1/3)^n-1? 

If you mean (1/3)⁸ = (-1/3)^n-1 then n = 9 is an integer solution.

For other non-integer solutions you need to involve complex numbers. 

If you mean (1/3)⁸ = (-1/3)^n-1 then n = 9 is an integer solution.

Please explain where the - sign in (-1/3) comes from to make lhs = rhs.

If n = 9 then (-1/3)^n-1 = (-1/3)⁸ = (-1)⁸/3⁸ = 1/3⁸ = 1⁸/3⁸ = (1/3)⁸