Problem:
If x=2−1+i3​​ and y=2−1−i3​​, where i2=−1, then which of the following is not correct?
Answer Choices:
A. x5+y5=−1
B. x7+y7=−1
C. x9+y9=−1
D. x11+y11=−1
E. x13+y13=−1
Solution:
Note that x3=y3=1, because x and y are the complex roots of 0=x3−1=(x−1)(x2+x+1). Alternatively, plot x and y in the complex plane and observe that they have modulus 1 and arguments 2Ï€/3 and 4Ï€/3. Thus x9+y9=1+1î€ =−1. (To show that the other equations are correct is easy if one also notes that y=x2 and x+y=−1. For instance, x11+y11=x2+x22=y+x=−1.)