Bài 33 trang 207 SGK giải tích 12 nâng cao

Đề bài

Bài 33. Tính \({\left( {\sqrt 3  - i} \right)^6};\,\,\,{\left( {{i \over {1 + i}}} \right)^{2004}};\,\,\,{\left( {{{5 + 3i\sqrt 3 } \over {1 - 2i\sqrt 3 }}} \right)^{21}}\)

Hướng dẫn giải

\({\left( {\sqrt 3  - i} \right)^6} = {\left[ {2\left( {\cos \left( { - {\pi  \over 6}} \right) + i\sin \left( { - {\pi  \over 6}} \right)} \right)} \right]^6} = {2^6}\left[ {\cos \left( { - \pi } \right) + i\sin \left( { - \pi } \right)} \right] =  - {2^6}\)

\({i \over {i + 1}} = {{1 + i} \over 2} = {1 \over {\sqrt 2 }}\left( {\cos {\pi  \over 4} + i\sin {\pi  \over 4}} \right)\) nên

\(\eqalign{  & {\left( {{1 \over {1 + i}}} \right)^{2004}} = {1 \over {{2^{1002}}}}\left( {\cos {{2004\pi } \over 4} + i\sin {{2004\pi } \over 4}} \right)  \cr  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {1 \over {{2^{1002}}}}\left( {\cos \pi  + i\sin \pi } \right) =  - {1 \over {{2^{1002}}}} \cr} \)

\({{5 + 3i\sqrt 3 } \over {1 - 2i\sqrt 3 }} = {{\left( {5 + 3i\sqrt 3 } \right)\left( {1 + 2i\sqrt 3 } \right)} \over {1 + 12}} = {{ - 13 + 13i\sqrt 3 } \over {13}} =  - 1 + i\sqrt 3 \)

\( = 2\left( { - {1 \over 2} + {{\sqrt 3 } \over 2}i} \right) = 2\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)\)

Do đó:

\({\left( {{{5 + 3i\sqrt 3 } \over {1 - 2i\sqrt 3 }}} \right)^{21}} = {2^{21}}\left( {\cos 14\pi  + i\sin 14\pi } \right) = {2^{21}}\)