Complex number exponent for wiki

$\cos\left(\dfrac{2\pi}{3}\right)=-1/2, \sin\left(\dfrac{2\pi}{3}\right)=\sqrt{3}/2$

Therefore given expression using Euler's form can be written as:

$\large{\begin{aligned}&\left(\cos\left(\color{#3D99F6}{\dfrac{2\pi}{3}}\right)+ i\sin\left(\color{#3D99F6}{\dfrac{2\pi}{3}}\right)\right)^3\\&=\left(e^{\color{#3D99F6}{\frac{2\pi i}{3}}}\right)^3\\&=e^{2\pi i}=\cos (2\pi)+i\sin(2\pi)=\boxed{1}\end{aligned}}$

Using complex number multiplication:

$\left(\frac{-1+i\sqrt{3}}{2}\right)^3=\frac{(-1+i\sqrt{3})^3}{2^3}=\frac{1}{8}\left(-1+i\sqrt{3}\right)^3$ $=\frac{1}{8}\left(-1+i\sqrt{3}\right)\left(-1+i\sqrt{3}\right)\left(-1+i\sqrt{3}\right)$ $=\frac{1}{8}\left(1-2i\sqrt{3}-3\right)\left(-1+i\sqrt{3}\right)=\frac{1}{8}\left(-2-2i\sqrt{3}\right)\left(-1+i\sqrt{3}\right)$ $=\frac{1}{8}\left(2-2i\sqrt{3}+2i\sqrt{3}+6\right)=\frac{1}{8}(8)=\boxed{1}$

This is clearly the cube root of unity. In maths use some rememberence as well so as to svore high in JEE type exams.

$(\frac{-1+i \sqrt{3})}{2}$ correspondes to $240°$ 2/3 of a full circle. Potentiating a complex number behaves as addition on the angles, leading $(\frac{-1+i \sqrt{3}}{2})^3$ corresponding to $720° = 360° = 0°$ with $\cos(0) = 1$ given the desired answer. For less text, more math and more accurrancy consider reading Rishabh Cool answer.