Q2014/005) If z= -2+ 2√3i , then find z ^(2n) + [2^(2n)][z^n]+ 2^(4n)?

We have  z= -2+ 2√3i = 4 cis 120 or 4 w where w is cube root of 1

So z ^(2n) + [2^(2n)][z^n]+ 2^(4n)

= 4^2n w^2n  + 4^2n w^n + 4^2n

= 4^2n ( w^2n + w ^n + 1)

= 16^n( w^2n + w ^n + 1)

Now there are 2 cases

1)      n is multiple of 3 :  so w^n = w^2n = 1 hence sum = 3 * 16^n

2)      n is not multiple of 3 : sum = 0