\[\boxed{\text{254}\text{\ (254)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[y = x^{n}\]
\[\textbf{а)}\ A\ (2;5):\ \ \ \]
\[y = 2^{n} \Longrightarrow функция\ возрастает;\]
\[2^{2} = 4;\ \ \ \ 2^{3} = 8;\ \ \ \]
\[4 < 5 < 8 \Longrightarrow\]
\[\Longrightarrow не\ существует\ такого\ \]
\[натурального\ n.\]
\[\textbf{б)}\ \text{B\ }\left( \sqrt{3};81 \right):\ \ \ \]
\[81 = \left( \sqrt{3} \right)^{n}\]
\[\left( \sqrt{3} \right)^{n} = \left( \left( \sqrt{3} \right)^{2} \right)^{4}\]
\[\left( \sqrt{3} \right)^{n} = \left( \sqrt{3} \right)^{8}\]
\[n = 8.\]
\[\textbf{в)}\ \text{C\ }( - 5;415):\ \ \ \]
\[415 = ( - 5)^{n} \Longrightarrow n - четное;\]
\[n = 2m;\ \ \ \]
\[y = ( - 5)^{2m} \Longrightarrow возрастает;\]
\[( - 5)^{2} = 25;\ \ \ ( - 5)^{4} = 625;\ \]
\[\ 25 < 415 < 625 \Longrightarrow\]
\[\Longrightarrow не\ существует\ такого\ n.\]
\[\textbf{г)}\ \text{D\ }( - 7;\ - 343):\ \ \ \]
\[- 343 = ( - 7)^{n}\]
\[( - 7)^{n} = ( - 7)^{3}\]
\[n = 3.\]