Explanation:
A \( \left[\left(\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}\right)^3\right]^{1 / 4}=(\cos \pi+\mathrm{i} \sin \pi)^{1 / 4}\)
\(=[\cos (2 \mathrm{k} \pi+\pi)+\mathrm{i} \sin (2 \mathrm{k} \pi+\pi)]^{1 / 4}, \mathrm{k}=0,1,2,3\)
\(=\cos (2 \mathrm{k}+1) \frac{\pi}{4}+\mathrm{i} \sin (2 \mathrm{k}+1) \frac{\pi}{4}, \mathrm{k}=0,1,2,3\)
The continued product of the four values is
\(\cos \left(\frac{\pi}{4}+\frac{3 \pi}{4}+\frac{5 \pi}{4}+\frac{7 \pi}{4}\right)+\mathrm{i} \sin \left(\frac{\pi}{4}+\frac{3 \pi}{4}+\frac{5 \pi}{4}+\frac{7 \pi}{4}\right)\)
\(=\cos 4 \pi+\mathrm{i} \sin 4 \pi=1+\mathrm{i} .0=1\)