1. **State the problem:** Simplify the expression $\sqrt{\frac{3}{20}}$. 2. **Recall the property of square roots:** $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ for $a,b > 0$. 3. **Apply the property:** $$\sqrt{\frac{3}{20}} = \frac{\sqrt{3}}{\sqrt{20}}$$ 4. **Simplify the denominator:** $$\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$$ 5. **Rewrite the expression:** $$\frac{\sqrt{3}}{2\sqrt{5}}$$ 6. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{5}$ to remove the radical from the denominator. $$\frac{\sqrt{3}}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3} \times \sqrt{5}}{2 \times \cancel{\sqrt{5}} \times \cancel{\sqrt{5}}} = \frac{\sqrt{15}}{2 \times 5}$$ 7. **Simplify denominator:** $$\frac{\sqrt{15}}{10}$$ **Final answer:** $$\boxed{\frac{\sqrt{15}}{10}}$$