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\begin{document}

\para{Key to worksheet}

\begin{enumerate}
\item  \begin{enumerate}
	\item  $2+2i = \sqrt{8}e^{i\pi/4}$
	\item  $-1+\sqrt{63}i = 8e^{(1.696)i}$
	\item  $3-3\sqrt{3}i = 6e^{-i\pi/3}$
	\item  $-3\sqrt{5}-2i = 6e^{(-2.8518)i}$
	\item  $-3+7i = \sqrt{58}e^{(1.9757)i}$
	\end{enumerate}

\item \begin{enumerate}
	\item  $5e^{i\pi} = -5$
	\item  $\sqrt{5}e^{(4.2487)i} = -1-2i$
	\end{enumerate}

\item \begin{enumerate}
	\item  $(-1+\sqrt{63}i)(2+2i) = -17.875 +13.875i = \sqrt{512} e^{(2.4815)i}$
	\item  $(-3\sqrt{5}-2i)(-3+7i) = 34.125-40.957i = \sqrt{2842}e^{(-.8762)i}$
	\item  $\frac{-1+\sqrt{63}i}{2+2i} = 1.734+2.234i = \sqrt{8}e^{(.9107)i}$
	\item  $\frac{-3\sqrt{5}-2i}{-3+7i} = .1056 +.9131i = \frac{7}{\sqrt{58}} e^{(1.4557)i}$
	\end{enumerate}

\item  \begin{enumerate}
	\item  $(2+2i)^5 = -128-128i = 128\sqrt{2} e^{-ie\pi/4}$
	\item  $(-1+\sqrt{63}i)^3 = 188-476.235i = 512 e^{(-1.1948)i}$
	\item  $(-3+7i)^{-3} = (0.002122) + (0.0007893)i = \frac{1}{58\sqrt{58}} e^{(.3561)i}$
	\end{enumerate}

\item  \begin{enumerate}
	\item  $5th$ roots of $1$: 
	\begin{eqnarray*}
	1 & = & e^{0i}  \\
	0.3090+0.9511i & = & e^{i2\pi/5} \\
	-0.8090+0.5878i & = & e^{i4\pi/5} \\
	-0.8090-0.5878i & = & e^{i6\pi/5} \\
	0.3090-0.9511i & = &  e^{i8\pi/5}
	\end{eqnarray*}

	\item  Cube roots of $2+2i$:
	\begin{eqnarray*}
	1.3660 + 0.3660i & = & \sqrt{2}e^{i\pi/12} \\
	-1+i & = & \sqrt{2}e^{i3\pi/4} \\
	-0.3660 - 1.3660i & = & \sqrt{2}e^{-i7\pi/12}
	\end{eqnarray*}

	\item  Square roots of $-3\sqrt{5}-2i$:
	\begin{eqnarray*}
	0.3820 - 2.6180i & = & \sqrt{7} e^{i(-1.426)} \\
	-0.3820 + 2.6180i & = & \sqrt{7} e^{i(1.7.16)}
	\end{eqnarray*}

	\item  Fouth roots of $-3+7i$:
	\begin{eqnarray*}
	1.4627 + 0.7876i & = & (58)^{1/8} e^{i(0.4939)} \\
	& = & (58)^{1/8} e^{i(2.0647)} \\
	& = & (58)^{1/8} e^{i(-2.6477)} \\
	& = & (58)^{1/8} e^{i(-1.0769)}
	\end{eqnarray*}

	\end{enumerate}
\end{enumerate}










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