\(\displaystyle x=\frac{3+\sqrt{13}}{2}\Rightarrow x^3-3x-1=0\Rightarrow x-x^{-1}=3\)
\(\Rightarrow \displaystyle x+x^{-1}=\sqrt{\left(x-x^{-1}\right)^2+4}=\sqrt{13}\)
\(\displaystyle \Rightarrow x^2+x^{-2}=\left(x-x^{-1}\right)^2+2=11\)
\(\displaystyle \Rightarrow x^3+x^{-3}=\left(x+x^{-1}\right)^3-3\cdot x\cdot x^{-1} \left(x+x^{-1}\right)=10\sqrt{13}\)
\(\displaystyle \Rightarrow x^5+x^{-5}=\left(x^3+x^{-3}\right)\left(x^2+x^{-2}\right)- \left(x+x^{-1}\right)=109\sqrt{13}\)
所求=\(\displaystyle \frac{x^{10}+x^8+x^2+1}{x^{10}+x^6+x^4+1}=\frac{x^5+x^3+x^{-3}+x^{-5}}{x^5+x+x^{-1}+x^{-5}}=\frac{119}{110}.\)