1. **State the problem:** Simplify the expression $$\sqrt{\frac{7x}{10y^3}}$$. 2. **Recall the property of square roots:** $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ for positive $a$ and $b$. 3. **Apply the property:** $$\sqrt{\frac{7x}{10y^3}} = \frac{\sqrt{7x}}{\sqrt{10y^3}}$$ 4. **Simplify the denominator:** $$\sqrt{10y^3} = \sqrt{10} \cdot \sqrt{y^3} = \sqrt{10} \cdot y^{\frac{3}{2}} = \sqrt{10} \cdot y^{1} \cdot y^{\frac{1}{2}} = y \sqrt{y}$$ 5. **Rewrite the expression:** $$\frac{\sqrt{7x}}{y \sqrt{10y}}$$ 6. **Final simplified form:** $$\boxed{\frac{\sqrt{7x}}{y \sqrt{10y}}}$$ This is the simplified form with the square root separated and the denominator expressed with $y$ outside the root and $\sqrt{y}$ inside.