1. **Problem:** Simplify the expression $$\sqrt{6561x^3}$$. 2. **Recall the rule:** The square root of a product is the product of the square roots: $$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$$. 3. **Step 1:** Factor inside the root: $$\sqrt{6561x^3} = \sqrt{6561} \times \sqrt{x^3}$$. 4. **Step 2:** Simplify $$\sqrt{6561}$$. Since $$6561 = 81^2$$, we have: $$\sqrt{6561} = 81$$. 5. **Step 3:** Simplify $$\sqrt{x^3}$$. Write $$x^3 = x^2 \times x$$, so: $$\sqrt{x^3} = \sqrt{x^2 \times x} = \sqrt{x^2} \times \sqrt{x} = x \sqrt{x}$$. 6. **Step 4:** Combine the results: $$\sqrt{6561x^3} = 81 \times x \sqrt{x} = 81x\sqrt{x}$$. **Final answer:** $$81x\sqrt{x}$$.