If $\text{ }\big(x-\frac{1}x\big)=i\sqrt{2}$. Then compute $\bigg(x^{2187}-\frac{1}{x^{2187}}\bigg)$. Here $i=\sqrt{-1}$
Solution 1:
$2187=3^7$. This is a clue. Powers of $3$ are significant. Now $$\left(x-\frac1x\right)^3=(i\sqrt2)^3=-2i\sqrt2$$ and $$\left(x-\frac1x\right)^3=x^3-\frac1{x^3}-3\left(x-\frac1x\right) =x^3-\frac1{x^3}-3i\sqrt2.$$ So $$x^3-\frac1{x^3}=i\sqrt2.$$ Repeating this, $$x^9-\frac1{x^9}=i\sqrt2,$$ $$x^{27}-\frac1{x^{27}}=i\sqrt2$$ etc. Eventually, $$x^{2187}-\frac1{x^{2187}}=i\sqrt2.$$