\[\begin{array}{l}{\text{Выполните действия в алгебраической форме}}{\text{.}}\\{\text{Результат запишите в тригонометрической и}}\\{\text{показательной формах}}{\text{.}}\\{\left( {\frac{{1 + i}}{{1 - i}}} \right)^{17}} + {i^6}\end{array}\]
\[\begin{array}{l}z = {\left( {\frac{{1 + i}}{{1 - i}}} \right)^{17}} + {i^6}\\\frac{{1 + i}}{{1 - i}} = \frac{{{{\left( {1 + i} \right)}^2}}}{{\left( {1 - i} \right)\left( {1 + i} \right)}} = \frac{{1 + 2i + {i^2}}}{{1 - {i^2}}} = \frac{{2i}}{2} = i\\{\left( {\frac{{1 + i}}{{1 - i}}} \right)^{17}} = {i^{17}} = {i^{16}} \cdot i = {\left( {{i^2}} \right)^8} \cdot i = {\left( { - 1} \right)^8} \cdot i = i\\{i^6} = {\left( {{i^2}} \right)^3} = {\left( { - 1} \right)^3} = - 1\\z = - 1 + i\\\left| z \right| = \sqrt {{{\left( { - 1} \right)}^2} + {1^2}} = \sqrt 2 \\{\mathop{\rm tg}\nolimits} \varphi = - 1\\\varphi = \frac{{3\pi }}{4}\\z = - 1 + i = \sqrt 2 \left( {\cos \frac{{3\pi }}{4} + i\sin \frac{{3\pi }}{4}} \right) = \sqrt 2 {e^{i\frac{{3\pi }}{4}}}\end{array}\]
\[z = - 1 + i = \sqrt 2 \left( {\cos \frac{{3\pi }}{4} + i\sin \frac{{3\pi }}{4}} \right) = \sqrt 2 {e^{i\frac{{3\pi }}{4}}}\]