1. **State the problem:** Simplify the expression $$\sqrt{19x^{13}}$$ assuming $$x > 0$$. 2. **Recall the property of radicals:** $$\sqrt{a b} = \sqrt{a} \times \sqrt{b}$$ and $$\sqrt{x^{2n}} = x^n$$ for $$x > 0$$. 3. **Rewrite the expression:** $$\sqrt{19x^{13}} = \sqrt{19} \times \sqrt{x^{13}}$$. 4. **Simplify $$\sqrt{x^{13}}$$:** Since $$13 = 2 \times 6 + 1$$, we can write $$x^{13} = x^{12} \times x^1 = (x^6)^2 \times x$$. 5. **Apply the square root:** $$\sqrt{x^{13}} = \sqrt{(x^6)^2 \times x} = \sqrt{(x^6)^2} \times \sqrt{x} = x^6 \sqrt{x}$$. 6. **Combine all parts:** $$\sqrt{19x^{13}} = \sqrt{19} \times x^6 \sqrt{x} = x^6 \sqrt{19x}$$. 7. **Final simplified form:** $$\boxed{x^6 \sqrt{19x}}$$. This matches the first option.