1. **State the problem:** Given $z = 4 \sqrt{3} e^{\frac{\pi i}{3}} - 4 e^{\frac{5\pi i}{6}}$, express $z$ in the form $re^{i\theta}$. Then show that $$\frac{z}{8} + i\left(\frac{z}{8}\right)^2 + \left(\frac{z}{8}\right)^3 = 2 e^{\frac{\pi i}{2}}.$$\n\n2. **Express $z$ in rectangular form:** Use Euler's formula $e^{i\alpha} = \cos \alpha + i \sin \alpha$.\n\nCalculate each term:\n$$4 \sqrt{3} e^{\frac{\pi i}{3}} = 4 \sqrt{3} \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) = 4 \sqrt{3} \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = 2 \sqrt{3} + 6 i.$$\n$$-4 e^{\frac{5\pi i}{6}} = -4 \left( \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} \right) = -4 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = 2 \sqrt{3} - 2 i.$$\n\n3. **Add the two parts:**\n$$z = (2 \sqrt{3} + 6 i) + (2 \sqrt{3} - 2 i) = 4 \sqrt{3} + 4 i.$$\n\n4. **Convert $z$ to polar form $re^{i\theta}$:**\nCalculate magnitude:\n$$r = |z| = \sqrt{(4 \sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8.$$\nCalculate argument:\n$$\theta = \tan^{-1} \left( \frac{4}{4 \sqrt{3}} \right) = \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) = \frac{\pi}{6}.$$\n\nThus,\n$$z = 8 e^{i \frac{\pi}{6}}.$$\n\n5. **Evaluate the expression:**\nLet $$w = \frac{z}{8} = e^{i \frac{\pi}{6}}.$$\nWe want to show:\n$$w + i w^2 + w^3 = 2 e^{i \frac{\pi}{2}}.$$\n\nCalculate each term:\n$$w = e^{i \frac{\pi}{6}},$$\n$$w^2 = e^{i \frac{\pi}{3}},$$\n$$w^3 = e^{i \frac{\pi}{2}}.$$\n\nSubstitute into the expression:\n$$w + i w^2 + w^3 = e^{i \frac{\pi}{6}} + i e^{i \frac{\pi}{3}} + e^{i \frac{\pi}{2}}.$$\n\nRewrite $i$ as $e^{i \frac{\pi}{2}}$:\n$$= e^{i \frac{\pi}{6}} + e^{i \frac{\pi}{2}} e^{i \frac{\pi}{3}} + e^{i \frac{\pi}{2}} = e^{i \frac{\pi}{6}} + e^{i \frac{5\pi}{6}} + e^{i \frac{\pi}{2}}.$$\n\nCalculate the sum of complex exponentials:\n$$e^{i \frac{\pi}{6}} = \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} = \frac{\sqrt{3}}{2} + i \frac{1}{2},$$\n$$e^{i \frac{5\pi}{6}} = \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} + i \frac{1}{2},$$\n$$e^{i \frac{\pi}{2}} = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = 0 + i \cdot 1 = i.$$\n\nSum real parts:\n$$\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} + 0 = 0.$$\nSum imaginary parts:\n$$\frac{1}{2} + \frac{1}{2} + 1 = 2.$$\n\nSo the sum is:\n$$0 + 2i = 2 e^{i \frac{\pi}{2}}.$$\n\n**Final answer:**\n$$z = 8 e^{i \frac{\pi}{6}},$$\nand\n$$\frac{z}{8} + i \left( \frac{z}{8} \right)^2 + \left( \frac{z}{8} \right)^3 = 2 e^{i \frac{\pi}{2}}.$$