2021 AMC 12A Problems/Problem 13

Problem

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, $\textbf{(B)} = 2\text{cis}(150), \textbf{(C)} =2\text{cis}(135)$ $, \textbf{(D)} =2\text{cis}(120)$ .

Taking the real part of the 5th power of each we have:

$\textbf{(A): }(-2)^5=-32$ ,

$\textbf{(B): }32\cos(650)=32\cos(30)=16\sqrt{3}$

$\textbf{(C): }32\cos(675)=32\cos(-45)=16\sqrt{2}$

$\textbf{(D): }32\cos(600)=32\cos(240)$ which is negative

$\textbf{(E): }(2i)^5$ which is imaginary

Thus, the answer is $\boxed{\textbf{(B)}}$ . ~JHawk0224

2021 AMC 12A (Problems • Answer Key • Resources)
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