Answer: $6^{300}<3^{500}<2^{800}<5^{400}$

$2^{800}=\left(2^8\right)^{100},\hspace{3mm} 3^{500}=\left(3^5\right)^{100},\hspace{3mm} 5^{400}=\left(5^4\right)^{100},\hspace{3mm} 6^{300}=\left(6^3\right)^{100}$

Put $2^{8},\hspace{2mm} 3^{5},\hspace{2mm} 5^{4},$ and $ 6^{3}$ in order of size.

$2^8=\left(2^4\right)^2=\left(\left(2^2\right)^2\right)^2=16^2=256$

$3^5=3^2\times3^3=9\times27=243$

$5^4=5^2\times5^2=25\times25=625$

$6^3=36\times6=216$

So $6^3<3^5<2^8<5^4$, so $6^{300}<3^{500}<2^{800}<5^{400}$.