Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part?
$\textbf{(A) }{-}2 \qquad \textbf{(B) }{-}\sqrt3+i \qquad \textbf{(C) }{-}\sqrt2+\sqrt2 i \qquad \textbf{(D) }{-}1+\sqrt3 i\qquad \textbf{(E) }2i$
First, $\textbf{(B)}$ is $2\text{cis}(150)$, $\textbf{(C)}$ is $2\text{cis}(135)$, $\textbf{(D)}$ is $2\text{cis}(120)$.
Taking the real part of the $5$th power of each we have:
$\textbf{(A): }(-2)^5=-32$
$\textbf{(B): }32\cos(750)=32\cos(30)=16\sqrt{3}$
$\textbf{(C): }32\cos(675)=32\cos(-45)=16\sqrt{2}$
$\textbf{(D): }32\cos(600)=32\cos(240)<0$
$\textbf{(E): }(2i)^5=32i$, whose real part is $0$
Thus, the answer is $\boxed{\textbf{(B) }{-}\sqrt3+i}$.
~JHawk0224