If x^2+x+1=0, then the value of (x+(1 / x))^4+(x^2+frac1x^2)^4+(x^3+frac1x^3)^4+ ... +(x^25+frac1x^25)^4 is :

x^2+x+1=0 => x=omega or omega^2 Therefore alpha=omega \ beta=omega^2 =(omega+omega^2)^4+(omega^2+omega^4)^4+(omega^3+omega^6)^4+ ... + (omega^25+omega^50)^4 =[(omega+omega^2)^4+(omega^2+omega^4)^4+(omega^4+omega^8)^4+ ... .+. .(omega^25+omega^50)^4]+[(omega^3+omega^6)^4+(omega^6+omega^12)^4+. .(omega^9+omega^18)^4+ ... +(omega^24+omega^48)^4] =underbrace[1+1+1 ... ... +1]_17 times +underbrace[(1+1)^4+(1+1)^4+ ... . .(1+1)^4]_8 times =17+128 =145

If x 2 x 1 0 then the value of x frac1x 4 x 2 frac1x 2 ... | null (Mathematics)

x^{2}+x+1=0\\

\Rightarrow \mathrm{x}=\omega

\omega^{2}\\

\therefore \alpha=\omega \& \beta=\omega^{2}\\

=\left(\omega+\omega^{2}\right)^{4}+\left(\omega^{2}+\omega^{4}\right)^{4}+\left(\omega^{3}+\omega^{6}\right)^{4}+\ldots+ \left(\omega^{25}+\omega^{50}\right)^{4}\\

=\left[\left(\omega+\omega^{2}\right)^{4}+\left(\omega^{2}+\omega^{4}\right)^{4}+\left(\omega^{4}+\omega^{8}\right)^{4}+\ldots .+\right. \left.\left(\omega^{25}+\omega^{50}\right)^{4}\right]+\left[\left(\omega^{3}+\omega^{6}\right)^{4}+\left(\omega^{6}+\omega^{12}\right)^{4}+\right. \left.\left(\omega^{9}+\omega^{18}\right)^{4}+\ldots+\left(\omega^{24}+\omega^{48}\right)^{4}\right]\\

=\underbrace{[1+1+1 \ldots \ldots+1]}_{17 \text { times }}+\underbrace{\left[(1+1)^{4}+(1+1)^{4}+\ldots . .(1+1)^{4}\right]}_{8 \text { times }}\\

=17+128\\

=145

Explanation

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