If `(1-i)^(2n)/((1+i)^(2n+1))` is real number then

Find the smallest positive integer value of n for which ((1+i)^n)/((1-i)^(n-2)) is a real number.

Find the smallest positive integer value of n for which ((1+i)^(n-2))/((1-i)^(n-2)) is a real number.

Find the smallest positive integer value of n for which ((1+i)^(n-2))/((1-i)^(n-2)) is a real number.

Find the smallest positive integer value of n for which ((1+i)^(n-2))/((1-i)^(n-2)) is a real number.

Statement - I : If n = 4m + 3 , is integer then i^(n) is equal to -i Statement- II : If n in N then (1 + i)^(2n) + (1- i)^(2n) is purely real number

The least positive integer n for which (1+i)^n/((1-i))^(n-2) is a real number is

(1+i)^(n_1)+(1+i^3)^(n_1)+(1+i^5)^(n_2)+(1+i^7)^(n_2) is a real number if

Show that , ((1+i)/(1-i))^2+((1-i)/(1+i))^2 is a real number.

For positive integers n_(1) and n_(2) the value of the expression (1+i)^(n_(1))+(1+i^(3))^(n_(1))+(1+i^(5))^(n_(2))+(1+i^(7))^(n_(2)) is a real number if and only if