1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} j^{\frac{1}{3}}$$. 2. **Recall the cube root and exponent rules:** - The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator: $$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$$. - When multiplying terms with the same root, you can combine them under one root if needed. - The expression $$j^{\frac{1}{3}}$$ means the cube root of $$j$$. 3. **Apply the cube root to the fraction:** $$\sqrt[3]{\frac{49}{27}} = \frac{\sqrt[3]{49}}{\sqrt[3]{27}}$$ 4. **Simplify the denominator:** $$\sqrt[3]{27} = 3$$ because $$3^3 = 27$$. 5. **Rewrite the expression:** $$\frac{\sqrt[3]{49}}{3} \times j^{\frac{1}{3}}$$ 6. **Combine the cube roots:** Since $$\sqrt[3]{49} = 49^{\frac{1}{3}}$$ and $$j^{\frac{1}{3}}$$, we can write: $$\frac{49^{\frac{1}{3}} j^{\frac{1}{3}}}{3} = \frac{(49j)^{\frac{1}{3}}}{3}$$ 7. **Final simplified form:** $$\boxed{\frac{\sqrt[3]{49j}}{3}}$$