d) \( \frac{\sqrt[3]{(a+b)^{-7}} \times(a+b)^{\frac{3}{3}}}{\sqrt[3]{7 p^{4} q^{7}}} \)

First, let's simplify the numerator step by step. The expression is: \[ \frac{\sqrt[3]{(a+b)^{-7}} \times (a+b)^{\frac{3}{3}}}{\sqrt[3]{7 p^{4} q^{7}}} \] First, notice that \( (a+b)^{\frac{3}{3}} = (a+b)^1 = (a+b) \). Now, we can combine the terms in the numerator: \[ \sqrt[3]{(a+b)^{-7}} \times (a+b) = (a+b)^{1 - \frac{7}{3}} \] Now, let’s simplify this exponent: \[ 1 - \frac{7}{3} = \frac{3}{3} - \frac{7}{3} = -\frac{4}{3} \] So, the numerator simplifies to: \[ (a+b)^{-\frac{4}{3}} \] Now we rewrite the whole expression: \[ \frac{(a+b)^{-\frac{4}{3}}}{\sqrt[3]{7 p^{4} q^{7}}} \] Next, we can express \( \sqrt[3]{7 p^{4} q^{7}} \) as: \[ (7 p^{4} q^{7})^{\frac{1}{3}} = 7^{\frac{1}{3}} p^{\frac{4}{3}} q^{\frac{7}{3}} \] At this point, the entire expression becomes: \[ \frac{(a+b)^{-\frac{4}{3}}}{7^{\frac{1}{3}} p^{\frac{4}{3}} q^{\frac{7}{3}}} \] We can write the final result clearly as: \[ \frac{(a+b)^{-\frac{4}{3}}}{7^{\frac{1}{3}} p^{\frac{4}{3}} q^{\frac{7}{3}}} \] This is the simplified form of the original expression.