1. **State the problem:** Simplify the expression $$S = \left(\left(-a^{3}\right)^{-\frac{2}{3}} - \left(\frac{\left(a^{-1}\right)^{3}a}{3} \middle/ \left(2a+1\right)^{\frac{5}{3}}\right)^{-\frac{5}{3}} + \frac{2}{a^{4}}\right)^{-\frac{1}{5}} \middle/ \left(\frac{1}{a^{7}} - \frac{1}{a^{10}}\right)^{-\frac{1}{5}}$$ 2. **Simplify each part step-by-step:** - Simplify $\left(-a^{3}\right)^{-\frac{2}{3}}$: $$\left(-a^{3}\right)^{-\frac{2}{3}} = \left(-1\right)^{-\frac{2}{3}} \cdot \left(a^{3}\right)^{-\frac{2}{3}} = (-1)^{-\frac{2}{3}} \cdot a^{-2}$$ Since $(-1)^{-\frac{2}{3}} = \frac{1}{(-1)^{\frac{2}{3}}} = \frac{1}{\left((-1)^2\right)^{\frac{1}{3}}} = \frac{1}{1} = 1$, so this equals: $$a^{-2}$$ - Simplify the fraction inside the second term: $$\frac{\left(a^{-1}\right)^{3}a}{3} = \frac{a^{-3} \cdot a}{3} = \frac{a^{-2}}{3}$$ - The denominator inside the parentheses is $\left(2a+1\right)^{\frac{5}{3}}$. - So the whole fraction inside the parentheses is: $$\frac{a^{-2}/3}{(2a+1)^{\frac{5}{3}}} = \frac{a^{-2}}{3(2a+1)^{\frac{5}{3}}}$$ - Now raise this to the power $-\frac{5}{3}$: $$\left(\frac{a^{-2}}{3(2a+1)^{\frac{5}{3}}}\right)^{-\frac{5}{3}} = \left(\frac{3(2a+1)^{\frac{5}{3}}}{a^{-2}}\right)^{\frac{5}{3}} = \left(3(2a+1)^{\frac{5}{3}} a^{2}\right)^{\frac{5}{3}}$$ - Distribute the exponent: $$3^{\frac{5}{3}} \cdot (2a+1)^{\frac{5}{3} \cdot \frac{5}{3}} \cdot a^{2 \cdot \frac{5}{3}} = 3^{\frac{5}{3}} (2a+1)^{\frac{25}{9}} a^{\frac{10}{3}}$$ - The third term inside the big parentheses is $\frac{2}{a^{4}} = 2a^{-4}$. 3. **Rewrite the big parentheses:** $$a^{-2} - 3^{\frac{5}{3}} (2a+1)^{\frac{25}{9}} a^{\frac{10}{3}} + 2a^{-4}$$ 4. **Now raise this entire expression to the power $-\frac{1}{5}$:** $$\left(a^{-2} - 3^{\frac{5}{3}} (2a+1)^{\frac{25}{9}} a^{\frac{10}{3}} + 2a^{-4}\right)^{-\frac{1}{5}}$$ 5. **Simplify the denominator:** $$\left(\frac{1}{a^{7}} - \frac{1}{a^{10}}\right)^{-\frac{1}{5}} = \left(a^{-7} - a^{-10}\right)^{-\frac{1}{5}} = \left(a^{-10}(a^{3} - 1)\right)^{-\frac{1}{5}}$$ 6. **Apply the exponent:** $$\left(a^{-10}(a^{3} - 1)\right)^{-\frac{1}{5}} = a^{2} (a^{3} - 1)^{-\frac{1}{5}}$$ 7. **Combine numerator and denominator:** $$S = \frac{\left(a^{-2} - 3^{\frac{5}{3}} (2a+1)^{\frac{25}{9}} a^{\frac{10}{3}} + 2a^{-4}\right)^{-\frac{1}{5}}}{a^{2} (a^{3} - 1)^{-\frac{1}{5}}} = \left(\frac{a^{-2} - 3^{\frac{5}{3}} (2a+1)^{\frac{25}{9}} a^{\frac{10}{3}} + 2a^{-4}}{a^{2} (a^{3} - 1)}\right)^{-\frac{1}{5}}$$ This is the simplified form of $S$. --- **Final answer:** $$S = \left(\frac{a^{-2} - 3^{\frac{5}{3}} (2a+1)^{\frac{25}{9}} a^{\frac{10}{3}} + 2a^{-4}}{a^{2} (a^{3} - 1)}\right)^{-\frac{1}{5}}$$