$\left(\frac{1+i\sqrt{3}}{2}\right)^{24}$

$\large \cos\dfrac{\pi}{3} +i \sin\dfrac{\pi}{3} = \ e ^{i \dfrac{\pi}{3}}$ .

$\large \implies \big( \cos\dfrac{\pi}{3} +i \sin\dfrac{\pi}{3} \big)^{24} =\ e^{24 i \frac{\pi}{3} }$

$\large = \ e^{8i \pi}$

$\large =\boxed{1}$