SOLUTION: Use De Moivres Theorem to show which integral powers of {{{ (-1+i)/(sqrt (2)) }}} are real, and which are imaginary.

Question 1118160: Use De Moivres Theorem to show which integral powers of are real, and which are imaginary.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Integral powers of are real
if the integer exponent is a multiple of 4 ( such as 0,4,8,..., or -4,-8, -12,...).
They are imaginary if the integer exponent is a multiple of 2, but not a multiple of 4
(such as 2,6,10,..., or -2,-6,-10,...).

If is an integer,

will be real if ,
and will be imaginary if .

if and only if for some integer ,
That means That the integral power is real only if and only if
is a multiple of 4.

if and only if for some integer ,
That means That the integral power is imaginary if and only if, for some integer ,
<--> ,
which means is even, but not a multiple of 4.
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