a) \(\frac{1}{9}\cdot3^4\cdot3^n=3^7\)
\(\Leftrightarrow\frac{1}{3^2}\cdot3^4\cdot3^n=3^7\)
\(\Leftrightarrow3^{n+2}=3^7\)
\(\Rightarrow n+2=7\)
\(\Rightarrow n=5\)
b) \(\left(2n+1\right)^3=343\)
\(\Leftrightarrow2n+1=7\)
\(\Leftrightarrow2n=6\)
\(\Rightarrow n=3\)
c) \(2\cdot16>2^n>4\)
\(\Leftrightarrow2^5>2^n>2^2\)
\(\Rightarrow5>n>2\)
d) \(n^{45}=n\)
\(\Leftrightarrow n^{45}-n=0\)
\(\Leftrightarrow n\left(n^{44}-1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}n=0\\n^{44}-1=0\end{cases}\Leftrightarrow}\orbr{\begin{cases}n=0\\n^{44}=1\end{cases}}\Leftrightarrow\orbr{\begin{cases}n=0\\n=\pm1\end{cases}}\)
e) \(\left(7n-11\right)^3=2^5\cdot5^2+200\)
\(\Leftrightarrow\left(7n-11\right)^3=1000\)
\(\Leftrightarrow7n-11=10\)
\(\Leftrightarrow7n=21\)
\(\Rightarrow n=3\)
