Problem:
If x=2β1+i3ββ and y=2β1βi3ββ, where i2=β1, then which of the following is not correct?
Answer Choices:
A. x5+y5=β1
B. x7+y7=β1
C. x9+y9=β1
D. x11+y11=β1
E. x13+y13=β1
Solution:
Note that x3=y3=1, because x and y are the complex roots of 0=x3β1=(xβ1)(x2+x+1). Alternatively, plot x and y in the complex plane and observe that they have modulus 1 and arguments 2Ο/3 and 4Ο/3. Thus x9+y9=1+1ξ =β1. (To show that the other equations are correct is easy if one also notes that y=x2 and x+y=β1. For instance, x11+y11=x2+x22=y+x=β1.)