1. **Stating the problem:** Simplify the expression $$\sqrt{\frac{45}{11x}}$$ and related radical expressions given. 2. **Recall the rule:** The square root of a fraction is the fraction of the square roots: $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$. 3. **Simplify the first expression:** $$\sqrt{\frac{45}{11x}} = \frac{\sqrt{45}}{\sqrt{11x}}$$ 4. **Simplify the numerator:** $$\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}$$ 5. **Rewrite the denominator:** $$\sqrt{11x} = \sqrt{11} \cdot \sqrt{x}$$ 6. **So the expression becomes:** $$\frac{3\sqrt{5}}{\sqrt{11} \cdot \sqrt{x}} = \frac{3\sqrt{5}}{\sqrt{11} \sqrt{x}}$$ 7. **For the second expression:** $$\sqrt{\frac{45}{\sqrt{11x}}} = \sqrt{45} \div \sqrt{\sqrt{11x}} = 3\sqrt{5} \div (11x)^{1/4}$$ 8. **For the third expression:** $$\frac{3 \sqrt{15 \cdot \sqrt{57}}}{11x}$$ Simplify inside the radical: $$15 \cdot \sqrt{57} = 15 \cdot 57^{1/2}$$ No further simplification without decimals. 9. **For the fourth expression:** $$\frac{3 \sqrt{55x}}{11x}$$ Rewrite numerator: $$3 \sqrt{55x} = 3 \sqrt{55} \sqrt{x}$$ 10. **Simplify the fraction by dividing numerator and denominator by $x$ if possible:** $$\frac{3 \sqrt{55} \sqrt{x}}{11x} = \frac{3 \sqrt{55} \cancel{\sqrt{x}}}{11 \cancel{x}}$$ But since $\sqrt{x} \neq x$, cannot cancel directly. **Final simplified form for the first expression:** $$\boxed{\frac{3\sqrt{5}}{\sqrt{11} \sqrt{x}}}$$