1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} \cdot j^{\frac{1}{3}}$$ where $j$ is a variable. 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: $$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$$ - Multiplying terms with the same root can be combined under one root: $$\sqrt[3]{x} \cdot \sqrt[3]{y} = \sqrt[3]{xy}$$ - Exponent rule: $$a^{m} \cdot a^{n} = a^{m+n}$$ 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 $27 = 3^3$ 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 $j^{\frac{1}{3}}$ is the cube root of $j$, multiply inside one cube root: $$\frac{\sqrt[3]{49} \cdot j^{\frac{1}{3}}}{3} = \frac{\sqrt[3]{49 \cdot j}}{3}$$ 7. **Final simplified form:** $$\boxed{\frac{\sqrt[3]{49j}}{3}}$$ This is the simplest exact form unless $j$ is known.