If α, β are the roots of the equations x^ 2 -2 x-1=0, then what is...

Since, (\alpha) and (\beta) are the roots of the equation $x^{2}-2 x-1=0$, then Sum of roots, (\alpha+\beta=2) and product of the roots (\alpha \beta=-1) Since, ((\alpha+\beta)=\alpha^{2}+\beta^{2}+2 \alpha \beta) (\Rightarrow 4=\alpha^{2}+\beta^{2}-2) (\Rightarrow \alpha^{2}+\beta^{2}=6) Now, (\alpha^{2} \beta^{-2}+\alpha^{-2} \beta^{2}=\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{\alpha^{4}+\beta^{4}}{(\alpha \beta)^{2}}$(\Rightarrow\left(\alpha^{2}+\beta^{2}\right)^{2}=$6^{2}) (\Rightarrow \alpha^{4}+\beta^{4}+2 \alpha^{2} \beta^{2}=36) (\Rightarrow \alpha^{4}+\beta^{4}+2=36) (\Rightarrow \alpha^{4}+\beta^{4}=34 \ldots..$(\mathrm{i})$ (\Rightarrow $\frac{\alpha^{4}+\beta^{4}}{(\alpha \beta)^{2}}=\frac{34}{(-1)^{2}}=$34) [Putting value of (\alpha^{4}+\beta^{4}=34) from Equation (i) $]$