[tex]\left(\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\right)^{2024}+\left(\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}\right)^{2024}= \\ \\ = \left(\cos\dfrac{\pi}{3}+i\sin \dfrac{\pi}{3}\right)^{2024}+\left[-\left(\cos\dfrac{2\pi}{3}+i\sin \dfrac{2\pi}{3}\right)\right]^{2024}\\ \\ = \left(\cos\dfrac{2024\pi}{3}+i\sin \dfrac{2024\pi}{3}\right)+\left(\cos\dfrac{4048\pi}{3}+i\sin \dfrac{4048\pi}{3}\right)[/tex]
[tex]=\cos\left(674\pi+\frac{2\pi}{3}\right)+i\sin\left(674\pi+\frac{2\pi}{3}\right) +\cos\left(1349\pi+\frac{\pi}{3}\right)+i\sin\left(1349\pi+\frac{\pi}{3}\right) \\ \\ =-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}-i\dfrac{\sqrt 3}{2}\\ \\ =\boxed{-1}[/tex]
