Task B-2.3. (20 points) Simplify the expression

$$
-x^{-x^{-x}}\left(\frac{x^{-x^{x}}+x^{x^{-x}}}{x^{-x^{-x}}+x^{x}}\right)
$$

for $x>0$.
Solution. By substituting $x^{x}=u$, we get:

$$
\begin{aligned}
& -x^{-x^{-x}}\left(\frac{x^{-x^{x}+x^{x-x}}}{x^{-x^{-x}}+x^{x^{x}}}\right)=-x^{-\frac{1}{u}} \cdot \frac{\frac{1}{x^{u}}+x^{\frac{1}{u}}}{\frac{1}{x^{\frac{1}{u}}}+x^{u}} \\
& =-x^{-\frac{1}{u}} \cdot \frac{\frac{1+x^{u} x^{\frac{1}{u}}}{x^{u}}}{\frac{1+x^{u} x^{\frac{1}{u}}}{x^{\frac{1}{u}}}} \\
& =-x^{-\frac{1}{u}} \cdot \frac{x^{\frac{1}{u}}}{x^{u}}=-\frac{1}{x^{u}}=-x^{-u}=-x^{-x^{x}}
\end{aligned}
$$

(20 points)