Simplificação de Expressão - CMF

\[
\begin{align*}
\frac{ \left( 4 + 2\sqrt{3} \right)^{2022} }{\left( 10 + 6 \sqrt{3} \right)^{1347} } & = \frac{2^{2022} \left( 2 + \sqrt{3} \right)^{2022} }{2^{1347} \left( 5 + 3 \sqrt{3} \right)^{1347} } \\
& = 2^{675} (2+\sqrt{3})^{675} \cdot \left[ \frac{2+ \sqrt{3}}{5 + 3\sqrt{3} } \right]^{1347} \\
& = 2^{675} (2+\sqrt{3})^{675} \cdot \frac{1}{(-2)^{1347}} [(2+\sqrt{3})(5 - 3\sqrt{3})]^{1347} \\
& = - 2^{675} (2+\sqrt{3})^{675} \cdot 2^{-1347} [1- \sqrt{3}]^{1347} \\
& = \frac{ \left( 2 +\sqrt{3} \right)^{675} \left( \sqrt{3} - 1 \right)^{1347} }{2^{672} }\\
& = \frac{ (\sqrt{3} + 1)^{1350} \left( \sqrt{3} - 1 \right)^{1347} }{2^{672 + 675} } \\
& = \frac{ 2^{1347} \left( \sqrt{3} + 1 \right)^3 }{2^{1347} }\\
& = \left( \sqrt{3}+1 \right)^3 \\
& = 3\sqrt{3} + 1 + 3\sqrt{3}( \sqrt{3} +1 ) \\
& = 10 + 6 \sqrt{3}
\end{align*}
\]
Assim,
\[
\begin{align*}
\frac{ \left( 4 + 2\sqrt{3} \right)^{2022} }{\left( 10 + 6 \sqrt{3} \right)^{1347} }+ 3 + 7\sqrt{3} & = 10 + 6 \sqrt{3} + 3 + 7\sqrt{3} \\
& = 13 + 13 \sqrt{3} \\
& = 13 (1 + \sqrt{3} )
\end{align*}
\]
Muito útil notar que
\[
2 + \sqrt{3} = \frac{4 + 2 \sqrt{3}}{2} = \frac{ \left(  \sqrt{3} +1\right)^2 }{2}
\]
Resposta: (E).