Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 653

\[1)\ \sqrt[4]{1}:\]

\[z^{4} = 1 = 1 + i \bullet 0 =\]

\[= \cos{2\pi n} + i\sin{2\pi n}\]

\[z = \cos\frac{\text{πn}}{2} + i\sin\frac{\text{πn}}{2}\]

\[z_{1} = \cos 0 + i\sin 0 = 1 + i \bullet 0 = 1;\]

\[z_{2} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = 0 + i \bullet 1 = i;\]

\[z_{3} = \cos\pi + i\sin\pi =\]

\[= - 1 + i \bullet 0 = - 1;\]

\[z_{4} = \cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2} =\]

\[= 0 + i \bullet ( - 1) = - i.\]

\[2)\ \sqrt[3]{- \frac{1}{27}}:\]

\[z^{3} = - \frac{1}{27} = \frac{1}{27}( - 1 + i \bullet 0) =\]

\[= \frac{1}{27}\left( \cos(\pi + 2\pi n) + i\sin(\pi + 2\pi n) \right)\]

\[z = \frac{1}{3}\left( \cos\frac{\pi + 2\pi n}{3} + i\sin\frac{\pi + 2\pi n}{3} \right)\]

\[z_{1} = \frac{1}{3}\left( \cos\frac{\pi}{3} + i\sin\frac{\pi}{3} \right) =\]

\[= \frac{1}{3}\left( \frac{1}{2} + \frac{\sqrt{3}}{2}i \right) = \frac{1}{6} + \frac{\sqrt{3}}{6}i;\]

\[z_{2} = \frac{1}{3}\left( \cos\pi + i\sin\pi \right) =\]

\[= \frac{1}{3}( - 1 + i \bullet 0) = - \frac{1}{3};\]

\[z_{3} = \frac{1}{3}\left( \cos\frac{5\pi}{3} + i\sin\frac{5\pi}{3} \right) =\]

\[= \frac{1}{3}\left( \frac{1}{2} - \frac{\sqrt{3}}{2}i \right) = \frac{1}{6} - \frac{\sqrt{3}}{6}i.\]

\[3)\ \sqrt[5]{1}:\]

\[z^{5} = 1 = 1 + i \bullet 0 =\]

\[= \cos{2\pi n} + i\sin{2\pi n}\]

\[z = \cos\frac{2\pi n}{5} + i\sin\frac{2\pi n}{5}\]

\[z_{1} = \cos 0 + i\sin 0 = 1 + i \bullet 0 = 1;\]

\[z_{2} = \cos\frac{2\pi}{5} + i\sin\frac{2\pi}{5};\]

\[z_{3} = \cos\frac{4\pi}{5} + i\sin\frac{4\pi}{5};\]

\[z_{4} = \cos\frac{6\pi}{5} + i\sin\frac{6\pi}{5};\]

\[z_{5} = \cos\frac{8\pi}{5} + i\sin\frac{8\pi}{5}.\]

\[4)\ \sqrt[4]{\sqrt{3} + i}:\]

\[z^{4} = \sqrt{3} + i = 2\left( \frac{\sqrt{3}}{2} + \frac{1}{2}i \right) =\]

\[= 2\left( \cos\left( \frac{\pi}{6} + 2\pi n \right) + i\sin\left( \frac{\pi}{6} + 2\pi n \right) \right)\]

\[z = \sqrt[4]{2}\left( \cos\left( \frac{\pi}{24} + \frac{\text{πn}}{2} \right) + i\sin\left( \frac{\pi}{24} + \frac{\text{πn}}{2} \right) \right)\]

\[z_{1} = \sqrt[4]{2}\left( \cos\frac{\pi}{24} + i\sin\frac{\pi}{24} \right);\]

\[z_{2} = \sqrt[4]{2}\left( \cos\frac{13\pi}{24} + i\sin\frac{13\pi}{24} \right);\]

\[z_{3} = \sqrt[4]{2}\left( \cos\frac{25\pi}{24} + i\sin\frac{25\pi}{24} \right);\]

\[z_{4} = \sqrt[4]{2}\left( \cos\frac{37\pi}{24} + i\sin\frac{37\pi}{24} \right).\]