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

Simplify the expression by following steps: - step0: Solution: \(\left(\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2}\div \left(\sqrt[3]{\frac{a^{-1}}{x^{-3}}}\right)\right)\) - step1: Remove the parentheses: \(\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2}\div \left(\sqrt[3]{\frac{a^{-1}}{x^{-3}}}\right)\) - step2: Rewrite the expression: \(\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2}\div \left(\sqrt[3]{\frac{x^{3}}{a}}\right)\) - step3: Simplify the root: \(\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2}\div \frac{x}{\sqrt[3]{a}}\) - step4: Rewrite the expression: \(\frac{x^{3}}{a^{\frac{10}{3}}}\div \frac{x}{\sqrt[3]{a}}\) - step5: Multiply by the reciprocal: \(\frac{x^{3}}{a^{\frac{10}{3}}}\times \frac{\sqrt[3]{a}}{x}\) - step6: Reduce the fraction: \(\frac{x^{2}}{a^{3}}\) Calculate or simplify the expression \( \left[\frac{\sqrt{a^{2}-x^{2}}}{a} \div\left(\frac{a^{3}+a^{2} x}{a-x}\right)\right] \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{\sqrt{a^{2}-x^{2}}}{a}\div \left(\frac{a^{3}+a^{2}x}{a-x}\right)\right)\) - step1: Remove the parentheses: \(\frac{\sqrt{a^{2}-x^{2}}}{a}\div \left(\frac{a^{3}+a^{2}x}{a-x}\right)\) - step2: Remove the parentheses: \(\frac{\sqrt{a^{2}-x^{2}}}{a}\div \frac{a^{3}+a^{2}x}{a-x}\) - step3: Multiply by the reciprocal: \(\frac{\sqrt{a^{2}-x^{2}}}{a}\times \frac{a-x}{a^{3}+a^{2}x}\) - step4: Multiply the terms: \(\frac{\sqrt{a^{2}-x^{2}}\times \left(a-x\right)}{a\left(a^{3}+a^{2}x\right)}\) - step5: Multiply the terms: \(\frac{\left(a-x\right)\sqrt{a^{2}-x^{2}}}{a\left(a^{3}+a^{2}x\right)}\) - step6: Simplify: \(\frac{\left(a-x\right)\sqrt{a^{2}-x^{2}}}{a^{4}+xa^{3}}\) - step7: Distribute \(\sqrt{a^{2}-x^{2}}:\) \(\frac{a\sqrt{a^{2}-x^{2}}-x\sqrt{a^{2}-x^{2}}}{a^{4}+xa^{3}}\) Let's simplify the given expression step by step. 1. **First Part:** \[ \left[\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2} \div\left(\sqrt[3]{\frac{a^{-1}}{x^{-3}}}\right)\right] \] - First, we simplify \(\left(\frac{x^{\frac{3}{2}}}{a^{\frac{5}{3}}}\right)^{2}\): \[ = \frac{x^{3}}{a^{\frac{10}{3}}} \] - Next, we simplify \(\sqrt[3]{\frac{a^{-1}}{x^{-3}}}\): \[ = \sqrt[3]{a^{-1}} \cdot \sqrt[3]{x^{3}} = \frac{1}{\sqrt[3]{a}} \cdot x = \frac{x}{\sqrt[3]{a}} \] - Now, we perform the division: \[ \frac{x^{3}}{a^{\frac{10}{3}}} \div \frac{x}{\sqrt[3]{a}} = \frac{x^{3}}{a^{\frac{10}{3}}} \cdot \frac{\sqrt[3]{a}}{x} = \frac{x^{2} \cdot \sqrt[3]{a}}{a^{\frac{10}{3}}} \] - This simplifies to: \[ = \frac{x^{2}}{a^{3}} \] 2. **Second Part:** \[ \left[\frac{\sqrt{a^{2}-x^{2}}}{a} \div\left(\frac{a^{3}+a^{2} x}{a-x}\right)\right] \] - We rewrite the division: \[ = \frac{\sqrt{a^{2}-x^{2}}}{a} \cdot \frac{a-x}{a^{3}+a^{2} x} \] - This gives us: \[ = \frac{(a-x)\sqrt{a^{2}-x^{2}}}{a(a^{3}+a^{2} x)} \] - Thus, the simplified form is: \[ = \frac{a\sqrt{a^{2}-x^{2}} - x\sqrt{a^{2}-x^{2}}}{a^{4}+xa^{3}} \] 3. **Final Expression:** Combining both parts, we have: \[ \frac{x^{2}}{a^{3}} + \frac{a\sqrt{a^{2}-x^{2}} - x\sqrt{a^{2}-x^{2}}}{a^{4}+xa^{3}} \] This is the simplified form of the original expression.