Answer
1
Work Step by Step
See p. 606,
If $z=r(\cos\theta+i\sin\theta)$, then for any integer $n$
$z^{n}=r^{n}(\cos n\theta+i\sin n\theta)$
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$z=-\displaystyle \frac{1}{2}-\frac{\sqrt{3}}{2}i$
$\theta$ terminates in quadrant III,
$r=\sqrt{\frac{1}{4}+\frac{3}{4}}=1,$
$\displaystyle \tan\theta=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} =\sqrt{3} \Rightarrow \displaystyle \theta=\frac{4\pi}{3}$.
$z=1(\displaystyle \cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3})$
$z^{15}=1^{15}(\displaystyle \cos(15\cdot\frac{4\pi}{3})+i\sin(15\cdot\frac{4\pi}{3}))$
$=\cos 20\pi+i\sin 20\pi$
$=1+0$
$=1$
