Two of the three values of ${\left(-1\right)}^{1/3}$ are

$\cos \left(\mathrm{\pi }/3\right)+\mathrm{isin}(\mathrm{\pi }/3), \cos (5\mathrm{\pi }/3)+\mathrm{isin}(5\mathrm{\pi }/3)$The third value is

$\begin{array}{rcl}{(-1)}^{1/3}& =& {\left(\cos \mathrm{\pi }+\mathrm{i} \sin \mathrm{\pi }\right)}^{1/3}\\ & =& \cos \frac{\mathrm{\pi }+2\mathrm{k\pi }}{3}+\mathrm{i} \sin \frac{\mathrm{\pi }+2\mathrm{k\pi }}{3} : \mathrm{k}=0, 1, 2\\ & =& \mathrm{cis} \frac{(2\mathrm{k}+1)\mathrm{\pi }}{3}; \mathrm{k}=0, 1, 2\\ & =& \mathrm{cis} \frac{\mathrm{\pi }}{3}, \mathrm{cis} \mathrm{\pi }, \mathrm{cis} \frac{5\mathrm{\pi }}{3}\end{array}$

$\therefore$ Third root is $\mathrm{cis} \mathrm{\pi }=\cos \mathrm{\pi }+\mathrm{i} \sin \mathrm{\pi }=-1$