1. Stating the problem: Simplify the expression $$\sqrt{\frac{18x^2}{z^6}}$$. 2. Recall the rule for square roots: $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ and $$\sqrt{x^2} = |x|$$. 3. Apply the rule to separate numerator and denominator: $$\sqrt{\frac{18x^2}{z^6}} = \frac{\sqrt{18x^2}}{\sqrt{z^6}}$$ 4. Simplify numerator and denominator separately: $$\sqrt{18x^2} = \sqrt{9 \cdot 2 \cdot x^2} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{x^2} = 3 \cdot \sqrt{2} \cdot |x|$$ $$\sqrt{z^6} = |z^{3}| = |z|^3$$ 5. Combine the results: $$\frac{3 \cdot \sqrt{2} \cdot |x|}{|z|^3}$$ 6. Final simplified form: $$\frac{3|x|\sqrt{2}}{|z|^3}$$