1. State the problem: Simplify the expression $\sqrt[3]{\frac{49}{27}}\, j^{1/3}$. 2. Use cube-root rules: For cube roots, $\sqrt[3]{a}\,\sqrt[3]{b}=\sqrt[3]{ab}$ and $\sqrt[3]{a}=\frac{\sqrt[3]{\text{numerator}}}{\sqrt[3]{\text{denominator}}}$. 3. Simplify $\sqrt[3]{\frac{49}{27}}$: 4. Rewrite $\frac{49}{27}$ as perfect cubes times leftovers: $49=7\cdot 7$ and $27=3\cdot 3\cdot 3$. 5. Evaluate cube roots: 6. Combine with $j^{1/3}$: 7. Final answer: $\sqrt[3]{\frac{49}{27}}\, j^{1/3}=\frac{\sqrt[3]{49}}{3}\, j^{1/3}$.