1. **State the problem:** Simplify the expression $$\sqrt{x}(x^{2n+1}) \cdot \sqrt[3]{x^{3n}}$$ where $$x > 0$$. 2. **Recall the rules:** - The square root $$\sqrt{x}$$ can be written as $$x^{\frac{1}{2}}$$. - The cube root $$\sqrt[3]{x^{3n}}$$ can be written as $$x^{\frac{3n}{3}} = x^n$$. - When multiplying powers with the same base, add the exponents: $$x^a \cdot x^b = x^{a+b}$$. 3. **Rewrite the expression using exponents:** $$\sqrt{x}(x^{2n+1}) \cdot \sqrt[3]{x^{3n}} = x^{\frac{1}{2}} \cdot x^{2n+1} \cdot x^n$$ 4. **Add the exponents:** $$x^{\frac{1}{2}} \cdot x^{2n+1} \cdot x^n = x^{\frac{1}{2} + 2n + 1 + n}$$ 5. **Simplify the exponent:** $$\frac{1}{2} + 2n + 1 + n = \frac{1}{2} + 3n + 1 = 3n + \frac{3}{2}$$ 6. **Final simplified expression:** $$x^{3n + \frac{3}{2}}$$ This is the simplified form of the original expression.