1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} \cdot j^{\frac{1}{3}}$$. 2. **Recall the formula:** The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator: $$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$$. 3. **Apply the formula:** $$\sqrt[3]{\frac{49}{27}} = \frac{\sqrt[3]{49}}{\sqrt[3]{27}}$$ 4. **Simplify the denominator:** $$\sqrt[3]{27} = 3$$ because $$3^3 = 27$$. 5. **Rewrite the expression:** $$\frac{\sqrt[3]{49}}{3} \cdot j^{\frac{1}{3}} = \frac{\sqrt[3]{49} \cdot j^{\frac{1}{3}}}{3}$$ 6. **Combine the cube roots:** Since $$\sqrt[3]{49} = 49^{\frac{1}{3}}$$ and $$j^{\frac{1}{3}}$$, we can write: $$\frac{49^{\frac{1}{3}} \cdot j^{\frac{1}{3}}}{3} = \frac{(49j)^{\frac{1}{3}}}{3}$$ 7. **Final simplified form:** $$\boxed{\frac{\sqrt[3]{49j}}{3}}$$ This is the simplest exact form of the expression.