Problem:
Of the following complex numbers z, which one has the property that z5 has the greatest real part?
Answer Choices:
A. β2
B. β3β+i
C. β2β+2βi
D. β1+3βi
E. 2i Solution:
Each of the five given numbers has the same modulus, 2. In polar form, the five numbers are, in the order of the answer choices, 2cis(Ο),2cis(65βΟ),2cis(43βΟ),2cis(32βΟ), and 2cis(21βΟ). Thus their fifth powers are, respectively,
32cis(5Ο)=32cis(Ο),32cis(625βΟ)=32cis(61βΟ),32cis(415βΟ)=32cis(β41βΟ),32cis(310βΟ)=32cis(β32βΟ), and 32cis(25βΟ)=32cis(21βΟ).β
All of the arguments have been rewritten here to lie between βΟ and Ο, inclusive. Therefore the one with the greatest real part is the one whose argument is closest to 0 . This is 32cis(61βΟ), the fifth power of β3β+iβ.