Simplify the following expression as much as possible:

$$
4 a \sqrt{\frac{(a-\sqrt{b})\left(a^{2}-b^{2}\right)}{a^{2}-b}} \cdot \frac{\sqrt{\frac{a+\sqrt{a^{2}-b}}{2}}+\sqrt{\frac{a-\sqrt{a^{2}-b}}{2}}}{\left(a+\sqrt{\frac{a-b}{a-b}}\right)^{2}-\left(a-\sqrt{\frac{a-b}{a+b}}\right)^{2}}
$$
Denoting the first factor by $A$, the numerator and the denominator of the second factor by $B$ and $C$ respectively,

$$
\begin{gathered}
A=4 a \sqrt{\frac{(a-\sqrt{b})(a+b)(a-b)}{(a+\sqrt{b})(a-\sqrt{b})}}=4 a \sqrt{\frac{(a+b)(a-b)}{a+\sqrt{b}}}, \\
B^{2}=\frac{a+\sqrt{a^{2}-b}}{2}+2 \sqrt{\frac{a^{2}-\left(a^{2}-b\right)}{4}}+\frac{a-\sqrt{a^{2}-b}}{2}=a+\sqrt{b}
\end{gathered}
$$

thus

$$
B=\sqrt{a+\sqrt{b}}
$$

and

$$
C=2 a \cdot 2 \sqrt{\frac{a-b}{a+b}}=4 a \sqrt{\frac{a-b}{a-b}}
$$

Therefore, our expression is

$$
A \frac{B}{C}=\frac{4 a \sqrt{a+b} \sqrt{a-b}}{\sqrt{a+\sqrt{b}}} \cdot \frac{\sqrt{a+\sqrt{b}} \cdot \sqrt{a+b}}{4 a \sqrt{a-b}}=a+b
$$

Bayer József (Bp. XX., Kossuth g. I. o. t.)