Notes
Problems
\(\textbf{1)}\) Find the fourth roots of \(16 \)
\(\,\,\,\,\, 2(\cos(0)+i\sin(0))=2, \)
\(\,\,\,\,\, 2(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))=2i, \)
\(\,\,\,\,\, 2(\cos(\pi)+i\sin(\pi))=-2, \)
\(\,\,\,\,\, 2(\cos(\frac{3\pi}{2})+i\sin(\frac{3\pi}{2}))=-2i \)
\(\textbf{2)}\) Find the fourth roots of \(8\sqrt{2}+8\sqrt{2}i \)
\(\,\,\,\,\, 2(\cos(\frac{\pi}{16})+i\sin(\frac{\pi}{16}))\approx 1.9616+0.3902i, \)
\(\,\,\,\,\, 2(\cos(\frac{9\pi}{16})+i\sin(\frac{9\pi}{16}))\approx -0.3902+1.9616i, \)
\(\,\,\,\,\, 2(\cos(\frac{17\pi}{16})+i\sin(\frac{17\pi}{16}))\approx -1.9616-0.3902i, \)
\(\,\,\,\,\, 2(\cos(\frac{25\pi}{16})+i\sin(\frac{25\pi}{16}))\approx 0.3902-1.9616i \)
\(\textbf{3)}\) Find the cube roots of \(8 \)
\( 2(\cos(0)+i\sin(0))=2, \)
\(2(\cos(\frac{2\pi}{3})+i\sin(\frac{2\pi}{3}))=-1+\sqrt{3}i,\)
\(2(\cos(\frac{4\pi}{3})+i\sin(\frac{4\pi}{3}))=-1-\sqrt{3}i\)
\(\textbf{4)}\) Find the 5th roots of \(16+16\sqrt{3}i \)
\(\,\,\,\,\, 2(\cos(\frac{\pi}{15})+i\sin(\frac{\pi}{15}))\approx 1.9563+0.4158i, \)
\(\,\,\,\,\, 2(\cos(\frac{7\pi}{15})+i\sin(\frac{7\pi}{15}))\approx 0.2091+1.9890i,\)
\(\,\,\,\,\, 2(\cos(\frac{13\pi}{15})+i\sin(\frac{13\pi}{15}))\approx -1.8271+0.8135i,\)
\(\,\,\,\,\, 2(\cos(\frac{19\pi}{15})+i\sin(\frac{19\pi}{15}))\approx -1.3383-1.4863i ,\)
\(\,\,\,\,\, 2(\cos(\frac{5\pi}{3})+i\sin(\frac{5\pi}{3}))\approx 1-1.7321i\)
\(\textbf{5)}\) Find the cube roots of \(1+i \)
\(\,\,\,\,\, \sqrt[6]{2}(\cos(\frac{\pi}{12})+i\sin(\frac{\pi}{12}))\approx 1.0842+0.2905i, \)
\(\,\,\,\,\, \sqrt[6]{2}(\cos(\frac{3\pi}{4})+i\sin(\frac{3\pi}{4}))\approx -0.7937+0.7937i,\)
\(\,\,\,\,\, \sqrt[6]{2}(\cos(\frac{17\pi}{12})+i\sin(\frac{17\pi}{12}))\approx -0.2905-1.0842i\)
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