A) \[1\]
B) \[0\]
C) \[{{\omega }^{2}}\]
D) \[\omega \]
Correct Answer: A
Solution :
Given, \[(1+\omega )\,(1+{{\omega }^{2}})(1+{{\omega }^{4}})(1+{{\omega }^{8}})\] \[=(1+\omega )(-\omega )\{1+{{\omega }^{2}}.\omega )(1+{{({{\omega }^{2}})}^{2}}.{{\omega }^{2}}\}\] \[[\because \,\,1+\omega +{{\omega }^{2}}=0]\] \[=(1+\omega )(-\omega )(1+\omega )(1+{{\omega }^{2}})\] \[[\because \,{{\omega }^{3}}=1]\] \[={{(1+\omega )}^{2}}(-\omega -{{\omega }^{3}})\] \[=(1+{{\omega }^{2}}+2\omega )(-\omega -{{\omega }^{3}})\] \[=(1+{{\omega }^{2}}+2\omega )(-\omega -1)\] \[=(-\omega +2\omega )(+{{\omega }^{2}})={{\omega }^{3}}=1\]
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