5) Simplify \( \left[\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2} \div\left(\sqrt[3]{\frac{a^{-1}}{x^{-3}}}\right)\right]+\left[\frac{\sqrt{a^{2}-x^{2}}}{a} \div\left(\frac{a^{3}+a^{2} x}{a-x}\right.\right. \)

To simplify the expression \( \left[\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2} \div\left(\sqrt[3]{\frac{a^{-1}}{x^{-3}}}\right)\right]+\left[\frac{\sqrt{a^{2}-x^{2}}}{a} \div\left(\frac{a^{3}+a^{2} x}{a-x}\right)\right] \), we can break it down step by step. First, take the first part: 1. The expression \( \left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2} \) simplifies to \( \frac{x^{3}}{a^{\frac{10}{3}}} \). 2. The cube root term \( \sqrt[3]{\frac{a^{-1}}{x^{-3}}} \) simplifies to \( \frac{\sqrt[3]{a^{-1}}}{\sqrt[3]{x^{-3}}} = \frac{1/a^{1/3}}{1/x} = \frac{x}{a^{1/3}} \). 3. Now you can rewrite the division as multiplication, giving you \( \frac{x^{3}}{a^{\frac{10}{3}}} \cdot \frac{a^{1/3}}{x} = \frac{x^{2} a^{1/3}}{a^{\frac{10}{3}}} = \frac{x^{2}}{a^{\frac{10}{3}-\frac{1}{3}}} = \frac{x^{2}}{a^{3}} \). Next, for the second part: 1. The expression \( \frac{\sqrt{a^{2}-x^{2}}}{a} \div \left(\frac{a^{3}+a^{2} x}{a-x}\right) \) can also be transformed. To divide, multiply by the reciprocal: \( \frac{\sqrt{a^{2}-x^{2}}}{a} \cdot \frac{a-x}{a^{3}+a^{2} x} \). 2. This leads to \( \frac{(a-x)\sqrt{a^{2}-x^{2}}}{a(a^{3} + a^{2} x)} \). Putting both parts together, the simplified form is: \[ \frac{x^{2}}{a^{3}} + \frac{(a-x)\sqrt{a^{2}-x^{2}}}{a(a^{3} + a^{2} x)} \] Now you have a beautiful simplified expression!