2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.

2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.

The expression is already in a mathematical form and does not require translation. However, if you need the problem statement in English, it would be:

2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.

Simplify the expression above.
## Solution.

Domain of definition: $x>1$.

$$
\begin{aligned}
& \frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}= \\
& =\frac{\sqrt{\frac{\sqrt{x-1}}{\sqrt{x+1}}+\frac{\sqrt{x+1}}{\sqrt{x-1}}-2} \cdot\left(x+1+\sqrt{x^{2}-1}+x-1\right)}{(\sqrt{x+1}-\sqrt{x-1})\left(\sqrt{(x+1)^{2}}+\sqrt{(x+1)(x-1)}+\sqrt{(x-1)^{2}}\right)}=
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0047.jpg?height=222&width=1110&top_left_y=1223&top_left_x=139)

$$
\begin{aligned}
& =\frac{\sqrt{\frac{(\sqrt{x+1}-\sqrt{x-1})^{2}}{\sqrt{x^{2}-1}}}}{\sqrt{x+1}-\sqrt{x-1}}=\sqrt{\frac{(\sqrt{x+1}-\sqrt{x-1})^{2}}{\sqrt{x^{2}-1} \cdot(\sqrt{x+1}-\sqrt{x-1})^{2}}}=\sqrt{\frac{1}{\sqrt{x^{2}-1}}}=\frac{1}{\sqrt[4]{x^{2}-1}} .
\end{aligned}
$$

Answer: $\frac{1}{\sqrt[4]{x^{2}-1}}$