d) \( \frac{\sqrt[3]{56 p^{7} q^{4}}}{\sqrt[3]{7 p^{4} q^{7}}} \)

To simplify the expression \( \frac{\sqrt[3]{56 p^{7} q^{4}}}{\sqrt[3]{7 p^{4} q^{7}}} \), we can combine the cube roots first: \[ \frac{\sqrt[3]{56 p^{7} q^{4}}}{\sqrt[3]{7 p^{4} q^{7}}} = \sqrt[3]{\frac{56 p^{7} q^{4}}{7 p^{4} q^{7}}} \] Now, we simplify the fraction inside the cube root: 1. **Simplify the constant:** \[ \frac{56}{7} = 8 \] 2. **Simplify the variable \( p \):** \[ \frac{p^{7}}{p^{4}} = p^{7-4} = p^{3} \] 3. **Simplify the variable \( q \):** \[ \frac{q^{4}}{q^{7}} = q^{4-7} = q^{-3} \] Now we can combine these results: \[ \frac{56 p^{7} q^{4}}{7 p^{4} q^{7}} = 8 p^{3} q^{-3} \] Now plug this back into the cube root: \[ \sqrt[3]{8 p^{3} q^{-3}} = \sqrt[3]{8} \cdot \sqrt[3]{p^{3}} \cdot \sqrt[3]{q^{-3}} \] Calculating each cube root: - \(\sqrt[3]{8} = 2\) - \(\sqrt[3]{p^{3}} = p\) - \(\sqrt[3]{q^{-3}} = q^{-1} = \frac{1}{q}\) Now, combine these results: \[ \sqrt[3]{8 p^{3} q^{-3}} = 2p \cdot \frac{1}{q} = \frac{2p}{q} \] Thus, the simplified expression is: \[ \frac{2p}{q} \]