1. **State the problem:** Simplify the expression $\sqrt{45x^{19}}$ assuming $x>0$. 2. **Recall the rule:** $\sqrt{a b} = \sqrt{a} \times \sqrt{b}$ and $\sqrt{x^{2n}} = x^n$ for $x>0$. 3. **Factor inside the square root:** $$\sqrt{45x^{19}} = \sqrt{45} \times \sqrt{x^{19}}$$ 4. **Simplify $\sqrt{45}$:** $$45 = 9 \times 5$$ $$\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$$ 5. **Simplify $\sqrt{x^{19}}$:** $$\sqrt{x^{19}} = x^{\frac{19}{2}} = x^{9 + \frac{1}{2}} = x^9 \sqrt{x}$$ 6. **Combine all parts:** $$\sqrt{45x^{19}} = 3\sqrt{5} \times x^9 \sqrt{x} = 3x^9 \sqrt{5x}$$ **Final answer:** $$\boxed{3x^9 \sqrt{5x}}$$