1. **State the problem:** Simplify the expression $$\frac{18 \sqrt[3]{135 a^{8}}}{6 \sqrt[3]{5 a^{2}}}$$. 2. **Write the expression clearly:** $$\frac{18 \sqrt[3]{135 a^{8}}}{6 \sqrt[3]{5 a^{2}}}$$. 3. **Simplify the coefficients outside the cube roots:** $$\frac{18}{6} = \cancel{\frac{18}{6}} = 3$$. 4. **Rewrite the cube roots as a single cube root of a fraction:** $$3 \times \frac{\sqrt[3]{135 a^{8}}}{\sqrt[3]{5 a^{2}}} = 3 \times \sqrt[3]{\frac{135 a^{8}}{5 a^{2}}}$$. 5. **Simplify inside the cube root:** $$\frac{135}{5} = 27$$ and $$\frac{a^{8}}{a^{2}} = a^{8-2} = a^{6}$$. So inside the cube root we have $$27 a^{6}$$. 6. **Rewrite the expression:** $$3 \times \sqrt[3]{27 a^{6}}$$. 7. **Use the property of cube roots:** $$\sqrt[3]{27} = 3$$ because $$3^{3} = 27$$. Also, $$\sqrt[3]{a^{6}} = a^{6/3} = a^{2}$$. 8. **Substitute back:** $$3 \times 3 \times a^{2} = 9 a^{2}$$. **Final answer:** $$9 a^{2}$$.