1. **State the problem:** Simplify the expression $$\sqrt{108a^6}$$ by removing all perfect squares from inside the square root, assuming $$a$$ is positive. 2. **Recall the rule:** $$\sqrt{xy} = \sqrt{x} \times \sqrt{y}$$ and $$\sqrt{a^{2n}} = a^n$$ for positive $$a$$. 3. **Factor the number inside the root:** $$108 = 36 \times 3$$, where 36 is a perfect square. 4. **Rewrite the expression:** $$\sqrt{108a^6} = \sqrt{36 \times 3 \times a^6} = \sqrt{36} \times \sqrt{3} \times \sqrt{a^6}$$. 5. **Simplify each square root:** $$\sqrt{36} = 6$$, and since $$a$$ is positive, $$\sqrt{a^6} = a^{\frac{6}{2}} = a^3$$. 6. **Combine the simplified parts:** $$6 \times a^3 \times \sqrt{3} = 6a^3\sqrt{3}$$. **Final answer:** $$\boxed{6a^3\sqrt{3}}$$