Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 799: 81

$\frac{\sqrt[5]{8}}{2}+\frac{\sqrt[5]{8}}{2}i,-0.49+0.95i,-1.06-0.17i,-0.17-1.06i,\text{ and }0.95-0.49i$.

For any complex number $z=r\left( \cos \theta +i\sin \theta \right)$, if $z\ne 0$, $n$ distinct complex roots in radians can be found by the formula given below: ${{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +2\pi \cdot k}{n} \right)+i\sin \left( \frac{\theta +2\pi \cdot k}{n} \right) \right]$ Where $k=0,1,2,3,....,n-1$. $z=-1-i$ can be rewritten as $z=\sqrt{2}\left( \cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4} \right)$. So, the fifth roots of $z=\sqrt{2}\left( \cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4} \right)$ are ${{z}_{k}}=\sqrt[5]{\sqrt{2}}\left[ \cos \left( \frac{225{}^\circ +360{}^\circ \cdot k}{5} \right)+i\sin \left( \frac{225{}^\circ +360{}^\circ \cdot k}{5} \right) \right],k=0,1,2,3,4$ So, we can find the five complex fifth roots as below: $\begin{align} & {{z}_{0}}=\sqrt[5]{\sqrt{2}}\left[ \cos \left( \frac{225{}^\circ +360{}^\circ \cdot 0}{5} \right)+i\sin \left( \frac{225{}^\circ +360{}^\circ \cdot 0}{5} \right) \right] \\ & =\sqrt[5]{\sqrt{2}}\left( \cos 45{}^\circ +i\sin 45{}^\circ \right) \\ & =\sqrt[5]{\sqrt{2}}\left( \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} \right) \\ & =\frac{\sqrt[5]{8}}{2}+\frac{\sqrt[5]{8}}{2}i \end{align}$ $\begin{align} & {{z}_{1}}=\sqrt[5]{\sqrt{2}}\left[ \cos \left( \frac{225{}^\circ +360{}^\circ \cdot 1}{5} \right)+i\sin \left( \frac{225{}^\circ +360{}^\circ \cdot 1}{5} \right) \right] \\ & =\sqrt[5]{\sqrt{2}}\left( \cos 117{}^\circ +i\sin 117{}^\circ \right) \end{align}$ Use calculator and get $\sqrt[5]{\sqrt{2}}\left( \cos 117{}^\circ +i\sin 117{}^\circ \right)\approx -0.49+0.95i$ $\begin{align} & {{z}_{2}}=\sqrt[5]{\sqrt{2}}\left[ \cos \left( \frac{225{}^\circ +360{}^\circ \cdot 2}{5} \right)+i\sin \left( \frac{225{}^\circ +360{}^\circ \cdot 2}{5} \right) \right] \\ & =\sqrt[5]{\sqrt{2}}\left( \cos 189{}^\circ +i\sin 189{}^\circ \right) \end{align}$ Use calculator and get $\sqrt[5]{\sqrt{2}}\left( \cos 189{}^\circ +i\sin 189{}^\circ \right)\approx -1.06-0.17i$ $\begin{align} & {{z}_{3}}=\sqrt[5]{\sqrt{2}}\left[ \cos \left( \frac{225{}^\circ +360{}^\circ \cdot 3}{5} \right)+i\sin \left( \frac{225{}^\circ +360{}^\circ \cdot 3}{5} \right) \right] \\ & =\sqrt[5]{\sqrt{2}}\left( \cos 261{}^\circ +i\sin 261{}^\circ \right) \end{align}$ Use calculator and get $\sqrt[5]{\sqrt{2}}\left( \cos 261{}^\circ +i\sin 261{}^\circ \right)\approx -0.17-1.06i$ $\begin{align} & {{z}_{4}}=\sqrt[5]{\sqrt{2}}\left[ \cos \left( \frac{225{}^\circ +360{}^\circ \cdot 4}{5} \right)+i\sin \left( \frac{225{}^\circ +360{}^\circ \cdot 4}{5} \right) \right] \\ & =\sqrt[5]{\sqrt{2}}\left( \cos 333{}^\circ +i\sin 333{}^\circ \right) \end{align}$ Use calculator and get $\sqrt[5]{\sqrt{2}}\left( \cos 333{}^\circ +i\sin 333{}^\circ \right)\approx 0.95-0.49i$ So, the fifth roots are $\frac{\sqrt[5]{8}}{2}+\frac{\sqrt[5]{8}}{2}i,\text{ }-0.49+0.95i,\text{ }-1.06-0.17i,\text{ }-0.17-1.06i,\text{ and }0.95-0.49i$. Hence, the fifth roots are $\frac{\sqrt[5]{8}}{2}+\frac{\sqrt[5]{8}}{2}i,\text{ }-0.49+0.95i,\text{ }-1.06-0.17i,\text{ }-0.17-1.06i$, and $0.95-0.49i$.