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Câu hỏi: Giá trị biểu thức $M={{\left( 1-i \right)}^{2020}}$ bằng
A. ${{2}^{1010}}$
B. $-{{2}^{1010}}$
C. ${{2}^{1010}}i$
D. $-{{2}^{1010}}i$
A. ${{2}^{1010}}$
B. $-{{2}^{1010}}$
C. ${{2}^{1010}}i$
D. $-{{2}^{1010}}i$
Ta có ${{\left( 1-i \right)}^{2020}}={{\left[ {{\left( 1-i \right)}^{2}} \right]}^{1010}}$.
Mà ${{\left( 1-i \right)}^{2}}=1-2i+{{i}^{2}}=1-2i-1=-2i$.
Vậy ${{\left( -2i \right)}^{1010}}={{2}^{1010}}{{i}^{1010}}={{2}^{1010}}{{\left( {{i}^{2}} \right)}^{505}}={{2}^{1010}}{{\left( -1 \right)}^{505}}=-{{2}^{1010}}$.
Mà ${{\left( 1-i \right)}^{2}}=1-2i+{{i}^{2}}=1-2i-1=-2i$.
Vậy ${{\left( -2i \right)}^{1010}}={{2}^{1010}}{{i}^{1010}}={{2}^{1010}}{{\left( {{i}^{2}} \right)}^{505}}={{2}^{1010}}{{\left( -1 \right)}^{505}}=-{{2}^{1010}}$.
Đáp án B.