$$\begin{gathered} \mathrm{Assertion} \left(\mathrm{A}\right) : {\left(\sqrt{3}+\mathrm{i}\right)}^6+{\left(\sqrt{3}-\mathrm{i}\right)}^6=-128 \\ \mathrm{Reason} \left(\mathrm{R}\right) : \mathrm{If} \mathrm{n} \mathrm{is} \mathrm{a} \mathrm{positive} \mathrm{integer} \mathrm{then} \\ {\left(\sqrt{3}+\mathrm{i}\right)}^{\mathrm{n}}+{\left(\sqrt{3}-\mathrm{i}\right)}^{\mathrm{n}}=2^{\mathrm{n}+1}\mathrm{Cos}\frac{\mathrm{n\pi }}{6} \end{gathered}$$

(A)

$\begin{array}{rcl} & & {\left(\sqrt{3}+\mathrm{i}\right)}^6+{\left(\sqrt{3}-\mathrm{i}\right)}^6\\ & =& 2^6\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)+2^6{\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)}^6\\ & =& 2^6\left({\left(\cos \frac{\mathrm{\pi }}{6}+\mathrm{i} \sin \frac{\mathrm{\pi }}{6}\right)}^6+{\left(\cos \frac{\mathrm{\pi }}{6}-\mathrm{i} \sin \frac{\mathrm{\pi }}{6}\right)}^6\right)\\ & =& 2^6\left(\mathrm{cos\pi }+\mathrm{i} \mathrm{sin\pi }+\mathrm{cos\pi }-\mathrm{i} \mathrm{sin\pi }\right)\\ & =& 2^6\left(2\mathrm{cos\pi }\right)\\ & =& 2^7(-1)=-2^7=-128\end{array}$

(R)

$\begin{array}{rcl} & & {\left(\sqrt{3}+\mathrm{i}\right)}^{\mathrm{n}}+{\left(\sqrt{3}-\mathrm{i}\right)}^{\mathrm{n}}\\ & =& 2^{\mathrm{n}}\left({\left(\cos \frac{\mathrm{\pi }}{6}+\mathrm{i} \sin \frac{\mathrm{\pi }}{6}\right)}^{\mathrm{n}}+{\left(\cos \frac{\mathrm{\pi }}{6}-\mathrm{i} \sin \frac{\mathrm{\pi }}{6}\right)}^{\mathrm{n}}\right)\\ & =& 2^{\mathrm{n}}\left(\cos \mathrm{n}\frac{\mathrm{\pi }}{6}+\mathrm{i} \sin \mathrm{n}\frac{\mathrm{\pi }}{6}+\cos \mathrm{n}\frac{\mathrm{\pi }}{6}-\mathrm{i} \sin \mathrm{n}\frac{\mathrm{\pi }}{6}\right)\\ & =& 2^{\mathrm{n}}\left(2\cos \mathrm{n}\frac{\mathrm{\pi }}{6}\right)=2^{\mathrm{n}+1} \cos \mathrm{n}\frac{\mathrm{\pi }}{6}\end{array}$