$$\begin{gathered} \mathrm{If} {\mathrm{x}}^2+\mathrm{x}+1=0, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \\ {\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)}^2+{\left({\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}\right)}^2+.....+{\left({\mathrm{x}}^{27}+\frac{1}{{\mathrm{x}}^{27}}\right)}^2 \mathrm{is} \end{gathered}$$

$\begin{array}{rcl}{\mathrm{x}}^2+\mathrm{x}+1& =& 0 \Rightarrow \mathrm{x}=\mathrm{w} \mathrm{or} {\mathrm{w}}^2\\ & & {\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)}^2+{\left({\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}\right)}^2+....+{\left({\mathrm{x}}^2+\frac{1}{27}\right)}^2\\ & =& \left( \mathrm{w}+\frac{1}{\mathrm{w}}\right)+{\left({\mathrm{w}}^2+\frac{1}{{\mathrm{w}}^2}\right)}^2+....{\left({\mathrm{w}}^{27}+\frac{1}{{\mathrm{w}}^{27}}\right)}^2\\ \mathrm{w}& =& {\mathrm{w}}^4={\mathrm{w}}^7=....={\mathrm{w}}^{25}\\ {\mathrm{w}}^2& =& {\mathrm{w}}^5={\mathrm{w}}^8=....={\mathrm{w}}^{26}\\ {\mathrm{w}}^3& =& {\mathrm{w}}^6={\mathrm{w}}^9=....{\mathrm{w}}^{24}={\mathrm{w}}^{27}\end{array}$

$\therefore$ Given expression 

$\begin{array}{rcl} & =& 9 \left({\left(\mathrm{w}+\frac{1}{\mathrm{w}}\right)}^2+{\left({\mathrm{w}}^2+\frac{1}{{\mathrm{w}}^2}\right)}^2+{\left({\mathrm{w}}^3+\frac{1}{{\mathrm{w}}^3}\right)}^2\right)\\ & =& 9\left({\left(\mathrm{w}+{\mathrm{w}}^2\right)}^2+\left({\mathrm{w}}^2+\mathrm{w}\right)+{\left(1+1\right)}^2\right)\\ & =& 9\left({\left(-1\right)}^2+{\left(-1\right)}^2+2^2\right)\\ & =& 9\left(1+1+4\right)\\ & =& 54\end{array}$