1. **State the problem:** Simplify the expression $3 \cdot \sqrt{\frac{1}{9}} \cdot \sqrt{3}$.\n\n2. **Recall the properties of square roots:** For any positive numbers $a$ and $b$, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. Also, $\sqrt{\frac{1}{9}} = \frac{1}{3}$ because $\sqrt{1} = 1$ and $\sqrt{9} = 3$.\n\n3. **Simplify $\sqrt{\frac{1}{9}}$:**\n$$\sqrt{\frac{1}{9}} = \frac{1}{3}$$\n\n4. **Rewrite the expression:**\n$$3 \cdot \frac{1}{3} \cdot \sqrt{3}$$\n\n5. **Cancel common factors:**\n$$\cancel{3} \cdot \frac{1}{\cancel{3}} \cdot \sqrt{3} = 1 \cdot \sqrt{3} = \sqrt{3}$$\n\n6. **Final answer:**\n$$\sqrt{3}$$