[ Identical Transformations ]

Knowing that $x^{2}+x+1=0$, determine $x^{14}+1 / x^{14}$.

#
Given $x < 0$. Dividing the given equation term by term by $x$, we get $x+1 / x=-1$. Multiplying the given equation term by term by $x: x^{3}+x^{2}+x=0$, from which $x^{3}=-\left(x^{2}+x\right)=1$ (due to our equation).

$$
\begin{aligned}
x^{14}+1 / x^{14} & =\left(x^{3}\right)^{4} x^{2}+1 /\left(x^{3}\right)^{4} x^{2}=x^{2}+1 / x^{2}= \\
& =(x+1 / x)^{2}-2=1-2=-1
\end{aligned}
$$

## Answer

-1 .