Example \ Given $x^{2}+x+1=0$, try to find the value of the rational expression $x^{14}+\frac{1}{x^{14}}$.
Solution: Clearly, $x \neq 1$. From $x^{2}+x+1=\frac{x^{3}-1}{x-1}=0$, we get $x^{3}-1=0$, which means $x$ is a cube root of unity not equal to 1. Therefore, $x^{14}+\frac{1}{x^{14}}=x^{3 \times 4} \times x^{2}+\frac{x}{x^{3 \times 5}}=x^{2}+x=-1$.