Chapter 9 - Polar Coordinates; Vectors - 9.3 The Complex Plane; De Moivre's Theorem - 9.3 Assess Your Understanding - Page 594: 45

$32i.$

Using De Moivre's Theorem ($(\cos{x}+i\sin{x})^n=\cos{nx}+i\sin{nx}.$) and a calculator: $[2(\cos{(\frac{\pi}{10})}+i(\sin{(\frac{\pi}{10}}))]^5=2^5(\cos{(5\cdot\frac{\pi}{10})}+i(\sin{(5\cdot\frac{\pi}{10})}))=32\cdot(\cos{(\frac{\pi}{2})}+i(\sin{(\frac{\pi}{2})}))=32\cdot(0+i)=32i.$