1. **State the problem:** Given the equation $x^{-1} = \frac{\sqrt{2}}{6}$, find the value of $x$. 2. **Recall the rule:** $x^{-1}$ means the reciprocal of $x$, so $x^{-1} = \frac{1}{x}$. 3. **Rewrite the equation:** $$\frac{1}{x} = \frac{\sqrt{2}}{6}$$ 4. **Solve for $x$ by taking the reciprocal of both sides:** $$x = \frac{1}{\frac{\sqrt{2}}{6}}$$ 5. **Simplify the complex fraction:** $$x = \frac{1}{\frac{\sqrt{2}}{6}} = \frac{1 \times 6}{\sqrt{2}} = \frac{6}{\sqrt{2}}$$ 6. **Rationalize the denominator:** $$x = \frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{2}$$ 7. **Simplify the fraction:** $$x = 3\sqrt{2}$$ **Final answer:** $$x = 3\sqrt{2}$$