Answer
$32i$
Work Step by Step
Recall: De Moivre's Theorem:
$$[r(\cos{x}+i\sin{x})]^a=r^a(\cos{(ax)}+\ i \sin{(ax)})$$
Apply the theorem above to obtain:
$[ 2(\cos (\frac{\pi}{10})+i\sin (\frac{\pi}{10})]^5
\\=2^5[\cos((5) (\frac{\pi}{10}) +i \sin((5)(\frac{\pi}{10}))]
\\=32[\cos(\frac{\pi}{2})+i(\sin(\frac{\pi}{2})]
\\=32 (0+\ i)
\\=32 i$
