Algebra

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\begin{array}{rcl}z&=&(1+\sqrt{3i})^3\r&=&2\varphi&=&\frac{\pi}{3}\z^3&=&2^3(cos(3\frac{\pi}{3})+sen\pi i)\z^3&=&8(-1+0i)\z^3&=&-8\end{array}
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\begin{array}{rcl}z&=&(-\sqrt{3}+i)^4\r&=&2\varphi&=&\frac{5}{6}\pi\z^4&=&2^4(cos4(\frac{5}{6}\pi)+sen(\frac{4}{3}\pi)i)\z^4&=&16(-0.5-0.866i)\z^4&=&-8-13.85i\end{array}
[pic]\begin{array}{rcl}z&=&(-3-3i)^5\r&=&\sqrt{18}\varphi&=&\frac{5}{4}\pi\z^5&=&(\sqrt{18})^5(cos5(\frac{5}{4}\pi)+sen(\frac{\pi}{4})i)\z^5&=&1374.61(0.707+0.707i)\z^5&=&971.85+971.85i\end{array}
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\begin{array}{rcl}z&=&(3-i)^3\r&=&\sqrt{10}\varphi&=&0.102\pi\z^3&=&(\sqrt{10})^3(cos3(0.102\pi)+sen(0.306\pi)i)\z^3&=&31.62(0.5724+0.8199i)\z^3&=&18.099+25.9252i\end{array}
[pic]\begin{array}{rcl}z&=&(1+\sqrt{3}i)^{\frac{1}3\r&=&2\varphi&=&\frac{\pi}{3}\z^{\frac{1}{3}}_1&=&2^{\frac{1}{3}}(cos(\frac{\frac{\pi}{3}+0}{3})+sen(\frac{\pi}{9})i)\z^{\frac{1}{3}}_1&=&2^{\frac{1}{3}(0.9396+0.342i)\z^{\frac{1}{3}}_1&=&1.1838+0.4055i\z^{\frac{1}{3}}_2&=&2^{\frac{1}{3}}(cos(\frac{\frac{\pi}{3}+2\pi}{3}})+sen(\frac{7}{9}\pi)i)\z^{\frac{1}{3}}_2&=&2^{\frac{1}{3}}(-0.766+0.6427i)\z^{\frac{1}{3}}_2&=&-0.965+0.8097i\z^{\frac{1}{3}}_3&=&2^{\frac{1}{3}}(cos(\frac{\frac{\pi}{3}+4\pi}{3})+sen(\frac{13}{9}\pi)i)\z^{\frac{1}{3}}_3&=&2^{\frac{1}{3}}(-0.1733-0.6548i)\z^{\frac{1}{3}}_3&=&-0.2179-1.2407i\end{array}
[pic]\begin{array}{rcl}z&=&(-\sqrt{3}+i)^{\frac{1}{4}}\r&=&2\varphi&=&\frac{5}{6}\pi\z_1&=&2^{\frac{1}{4}}(cos(\frac{\frac{5}{6}\pi+0}{4})+sen(\frac{5}{24}\pi)i)\z_1&=&2^{\frec{1}{4}(0.7933+0.6087i)\z_1&=&0.9433+0.7238i\z_2&=&2^{\frac{1}{4}(cos(\frac{\frac{5}{6}\pi+2\pi}{4})+sen(\frac{17}{24}\pi)i)\z_2&=&2^{\frac{1}{4}}(-0.6087+0.7933i)\z_2&=&-0.7238+0.9433i\z_3&=&2^{\frac{1}{4}}(cos(\frac{\frac{5}{6}\pi+4\pi}{4})+sen(\frac{29}{24}\pi)i)\z_3&=&2^{\frac{1}{4}}(-0.7933-0.6087i)\z_3&=&-0.9433-0.7238i\z_4&=&2^{\frac{1}{4}}(cos(\frac{\frac{5}{6}\pi+6\pi}{4})+sen(\frac{41}{24}\pi)i)\z_4&=&2^{\frac{1}{4}}(0.6087-0.7933i)\z_4&=&0.7233-0.9433i\end{array}
[pic]\begin{array}{rcl}z&=&(-3-3i)^{\frac{1}{3}}\r&=&\sqrt{18}\varphi&=&\frac{5}{4}\pi\z_1&=&18^{\frac{1}{5}}(cos\frac{\frac{5}{4}\pi+0}{3})+sen(\frac{5}{12}\pi)i)\z_1&=&18^{\frac{1}{5}}(0.2588+0.9659i)\z_1&=&0.4613+1.7218i\z_2&=&18^{\frac{1}{5}}(cos(\frac{\frac{5}{4}\pi+2\pi}{3})+sen(\frac{13}{12}\pi)i)\z_2&=&18^{\frac{1}{5}}(-0.9659-0.2588i)\z_2&=&-1.7218-0.4613i\z_3&=&18^{\frac{1}{5}}(cos(\frac{\frac{5}{4}\pi+4\pi}{3})+sen(\frac{21}{12}\pi)i)\z_3&=&18^{\frac{1}{5}}(0.707-0.707i)\z_3&=&1.2602-1.2602i\end{array}[pic]\begin{array}{rcl}z&=&(3-i)^{\frac{1}{4}}\r&=&\sqrt{10}\varphi&=&0.102\pi\z_1&=&10^{\frac{1}{8}}(cos(\frac{0.102\pi+0}{4})+sen(0.0255\pi)i)\z_1&=&10^{\frac{1}{8}}(0.9967+0.08i)\z_1&=&1.3291+0.106i\z_2&=&10^{\frac{1}{8}}(cos(\frac{0.102\pi+2\pi}{4})+sen(0.5255\pi)i)\z_2&=&10^{\frac{1}{8}}(-0.08+0.9967i)\z_2&=&-0.106+1.3291i\z_3&=&10^{\frac{1}{8}}(cos(\frac{0.102\pi+4\pi}{4})+sen(1.0255\pi)i)\z_3&=&10^{\frac{1}{8}}(-0.9967-0.08i)\z_3&=&-1.3291-0.106i\z_4&=&10^{\frac{1}{8}}(cos(\frac{0.102\pi+6\pi}{4})+sen(1.5255\pi)i)\z_4&=&10^{\frac{1}{8}}(0.08-0.9967i)\z_4&=&0.106-1.3291i\end{array}[pic]\begin{array}{rcl}z&=&4^{\frac{1}{4}}\r&=&4\varphi&=&0\z_1&=&4^{\frac{1}{4}}(cos(\frac{0+0}{4})+sen(0)i)\z_1&=&4^{\frac{1}{4}}(1+0i)\z_1&=&1.414\z_2&=&4^{\frac{1}{4}}(cos(\frac{0+2\pi}{4})+sen{\frac{\pi}{2})i)\z_2&=&4^{\frac{1}{4}}(0+i)\z_2&=&1.414i\z_3&=&4^{\frac{1}{4}}(cos(\frac{0+4\pi}{4})+sen(\pi)i)\z_3&=&4^{\frac{1}{4}}(-1+0i)\z_3&=&-1.414\z_4&=&4^{\frac{1}{4}}(cos(\frac{0+6\pi}{4})+sen(\frac{3}{2}\pi)i)\z_4&=&4^{\frac{1}{4}}(0-i)\z_4&=&-1.414i\end{array}
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