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Question Number 148155 by Khalmohmmad last updated on 25/Jul/21

Answered by Rasheed.Sindhi last updated on 26/Jul/21

$$x+\frac{1}{x}=-1\Rightarrow x^2+x+1=0$$$$x=\omega ,{\omega }^2$$$$x=\omega$$$$x^{1400}+x^{1363}+x^{37}$$$$={\omega }^{1400}+{\omega }^{1363}+{\omega }^{37}$$$$={\omega }^{1400mod3}+{\omega }^{1363mod3}+{\omega }^{37mod3}$$$$={\omega }^2+\omega +\omega =-1+\omega$$$$x={\omega }^2$$$$={\left({\omega }^2\right)}^2+{\omega }^2+{\omega }^2$$$$=\omega .{\omega }^3+{\omega }^2+{\omega }^2$$$$=\omega +{\omega }^2+{\omega }^2$$$$=-1+{\omega }^2$$

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