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?

$\textbf{(A) }-2 \qquad \textbf{(B) }-\sqrt3+i \qquad \textbf{(C) }-\sqrt2+\sqrt2 i \qquad \textbf{(D) }-1+\sqrt3 i\qquad \textbf{(E) }2i$

Solution

$\textbf{(B)} = 2\text{cis}(150)$ $\textbf{(C)} =2\text{cis}(135)$ $\textbf{(D)} =2\text{cis}(120)$ 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)
Preceded by Problem 12	Followed by Problem 14
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All AMC 12 Problems and Solutions

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2021 AMC 12A Problems/Problem 13

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