4. Simplify the algebraic expression:

$$
\frac{\left(m^{2}-\frac{1}{n^{2}}\right)^{m} \cdot\left(n+\frac{1}{m}\right)^{n-m}}{\left(n^{2}-\frac{1}{m^{2}}\right)^{n}\left(m-\frac{1}{n}\right)^{m-n}}
$$
Solution. Let's denote the given expression by J. We have:

$$
\begin{aligned}
J & =\frac{\left(m^{2} n^{2}-1\right)^{m} \cdot(m n+1)^{n-m} \cdot m^{2 n} \cdot n^{m-n}}{\left(m^{2} n^{2}-1\right)^{n} \cdot(m n-1)^{m-n} \cdot n^{2 m} \cdot m^{n-m}} \\
& =\frac{\left(m^{2} n^{2}-1\right)^{m-n}}{(m n-1)^{m-n}} \cdot \frac{(m n+1)^{n-m} \cdot m^{2 n-n+m}}{n^{2 m-m+n}} \\
& =\frac{(m n-1)^{m-n}}{(m n-1)^{m-n}} \frac{(m n+1)^{m-n}}{(m n+1)^{m-n}} \frac{m^{m+n}}{n^{m+n}}=\left(\frac{m}{n}\right)^{m+n}
\end{aligned}
$$

## 2nd year