A) \[128\,\omega \]
B) \[-128\,\omega \]
C) \[128\,{{\omega }^{2}}\]
D) \[-128\,{{\omega }^{2}}\]
Correct Answer: D
Solution :
If \[\omega \] is a cube root of unity, then \[1+\omega +{{\omega }^{2}}=0\] and \[{{\omega }^{3}}=1\]. Now, \[{{(1+\omega -{{\omega }^{2}})}^{7}}={{(-{{\omega }^{2}}-{{\omega }^{2}})}^{7}}\] \[(\because 1+\omega +{{\omega }^{2}}=0)\] \[={{(-2{{\omega }^{2}})}^{7}}=-{{2}^{7}}.\,{{\omega }^{14}}\] \[=-128\,{{({{\omega }^{3}})}^{4}}{{\omega }^{2}}=-128\,{{\omega }^{2}}\] \[(\because {{\omega }^{3}}=1)\]
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