According to wolframalpha:
$\exp(-2\pi i/3)(-\sqrt{3}-i)^{1/3}$ is equal to $-(\sqrt{3}+i)^{1/3}$.
How to prove it or what is the general approach to convert the trigonometric functions (like LHS of the topic) to radical?
2026-03-27 21:44:21.1774647861
Trigonometric function to radicals conversion
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The left-hand side represents three distinct numbers that cubed will give $$ (\exp(-2\pi i/3))^3(-\sqrt{3}-i)=\exp(-2\pi i)(-\sqrt{3}-i)=-(\sqrt{3}+i) $$ so you have your claim.