Chapter 9 - Polar Coordinates; Vectors - 9.3 The Complex Plane; De Moivre's Theorem - 9.3 Assess Your Understanding - Page 594: 48

$\frac{1}{32}$

Using De Moivre's Theorem ($(\cos{x}+i\sin{x})^n=\cos{nx}+i\sin{nx}.$): $[0.5(\cos{(72^\circ)}+i(\sin{(72^\circ}))]^5=0.5^5(\cos{(5\cdot72^\circ)}+i(\sin{(5\cdot72^\circ})))=\frac{1}{32}\cdot(\cos{(360^\circ)}+i(\sin{(360^\circ})))=\frac{1}{32}\cdot(1+0)=\frac{1}{32}.$