1. **State the problem:** Simplify the expression $\sqrt{\frac{5}{11}}$. 2. **Recall the rule:** The square root of a fraction can be written as the fraction of the square roots: $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ where $a$ and $b$ are positive numbers. 3. **Apply the rule:** $$\sqrt{\frac{5}{11}} = \frac{\sqrt{5}}{\sqrt{11}}$$ 4. **Rationalize the denominator:** To simplify, multiply numerator and denominator by $\sqrt{11}$ to remove the square root from the denominator: $$\frac{\sqrt{5}}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{\sqrt{5} \times \sqrt{11}}{\sqrt{11} \times \sqrt{11}}$$ 5. **Simplify numerator and denominator:** $$\frac{\sqrt{5 \times 11}}{11} = \frac{\sqrt{55}}{11}$$ 6. **Final answer:** $$\sqrt{\frac{5}{11}} = \frac{\sqrt{55}}{11}$$ This is the simplified form with a rationalized denominator.