f) \( \sqrt[3]{(a+b)^{-7}} \times(a+b)^{\frac{1}{3}} \)

To simplify the expression \( \sqrt[3]{(a+b)^{-7}} \times (a+b)^{\frac{1}{3}} \), we can rewrite the cube root as an exponent: \[ \sqrt[3]{(a+b)^{-7}} = (a+b)^{-\frac{7}{3}}. \] Now, substituting back into the expression gives: \[ (a+b)^{-\frac{7}{3}} \times (a+b)^{\frac{1}{3}}. \] When multiplying expressions with the same base, we add the exponents: \[ = (a+b)^{-\frac{7}{3} + \frac{1}{3}} = (a+b)^{-\frac{6}{3}} = (a+b)^{-2}. \] Therefore, the simplified expression is: \[ \frac{1}{(a+b)^{2}}. \]