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) \(]\)