If ${\mathrm{x}}^2+\mathrm{x}+1=0$. then the value of

$\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)+\left({\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}\right)+\dots \dots +\left({\mathrm{x}}^{52}+\frac{1}{{\mathrm{x}}^{52}}\right)$ equals

$\begin{array}{l}If{x}^2+x+1=0thenx=\omega ,{\omega }^2\\ \therefore (x+\frac{1}{x})+(x^2+\frac{1}{x^2})+---+(x^{52}+\frac{1}{x^{52}})\\ =(\omega +\frac{1}{\omega })+({\omega }^2+\frac{1}{{\omega }^2})+---+({\omega }^{52}+\frac{1}{{\omega }^{52}})\\ =(\omega +{\omega }^2)+({\omega }^2+w)+---+(\omega +{\omega }^2)\\ =(-1)+(-1)+---+(-1)\left(52times\right)\\ =-52\end{array}$