1. **State the problem:** Simplify the expression $$\frac{\sqrt{80x^3}}{\sqrt{5x}}$$ using the quotient rule for square roots, assuming $x > 0$. 2. **Recall the quotient rule for square roots:** $$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$ for $a,b > 0$. 3. **Apply the quotient rule:** $$\frac{\sqrt{80x^3}}{\sqrt{5x}} = \sqrt{\frac{80x^3}{5x}}$$ 4. **Simplify inside the square root:** $$\frac{80x^3}{5x} = \frac{\cancel{80}^{{16}}x^{\cancel{3}}^{2}}{\cancel{5}^1x^{\cancel{1}}^{0}} = 16x^2$$ 5. **Rewrite the expression:** $$\sqrt{16x^2}$$ 6. **Simplify the square root:** Since $x > 0$, $$\sqrt{16x^2} = \sqrt{16} \cdot \sqrt{x^2} = 4x$$ **Final answer:** $$4x$$