\(\frac{(a - \frac{1}{a}) (a^{\frac{4}{3}} + \frac{1}{a^{\frac{2}{3}}})}{a^2 - \left(\frac{1}{a}\right)^2}\) \( = \frac{(a - \frac{1}{a}) \left(a^{\frac{4}{3}} + \frac{1}{a^{\frac{2}{3}}}\right)}{\left(a - \frac{1}{a}\right)\left(a + \frac{1}{a}\right)} \) \(= \frac{a^{\frac{4}{3}} + \frac{1}{a^{\frac{2}{3}}}}{a + \frac{1}{a}}\) \(= \frac{\frac{a^{2} + 1}{a^{\frac{2}{3}}}}{\frac{a^2 + 1}{a}}\) \(= \frac{1}{a^{\frac{2}{3}}} \cdot a\) \(= a^{1 - \frac{2}{3}} = a^{\frac{1}{3}}\) \(\boxed{a^{\frac{1}{3}}}\)