Let's rationalize the denominator of the following fraction and simplify the resulting expression to its simplest form:

$$
\frac{\sqrt{3+\sqrt{5}}}{\sqrt[3]{\left(4 \sqrt{2}-2 \sqrt{10)}^{2}\right.}}
$$
$$
\begin{aligned}
& \frac{\sqrt{3+\sqrt{5}}}{\sqrt[3]{\left(4 \sqrt{2}-2 \sqrt{10}\right)^{2}}}=\frac{\sqrt{3+\sqrt{5}} \sqrt[3]{4 \sqrt{2}-2 \sqrt{10}}}{4 \sqrt{2}-2 \sqrt{10}}= \\
& =\frac{\sqrt{3+\sqrt{5}} \sqrt[3]{4 \sqrt{2}-2 \sqrt{10}}(4 \sqrt{2}+2 \sqrt{10})}{32-40}= \\
& =-\frac{\sqrt[6]{(3+\sqrt{5})^{3}(4 \sqrt{2}-2 \sqrt{10})^{2}} \cdot 2 \cdot \sqrt{2}(2-\sqrt{5})}{8}= \\
& =-\frac{\sqrt[6]{8(3+\sqrt{5})^{3}(2-\sqrt{5})^{2}}(2-\sqrt{5}) \sqrt{2}}{4}= \\
& =-\frac{\sqrt[6]{64(9+4 \sqrt{5})(9-4 \sqrt{5})}(2-\sqrt{5}) \sqrt{2}}{4}=\frac{\sqrt{2}}{2}(\sqrt{5}-2)
\end{aligned}
$$

(Vilmos Erdős, Budapest.)

The problem was also solved by: Bauer E., Dénes M., Grün E., Koffler B., Lengyel K., Lengyel P., Neumann L., Sichermann F., Steiner L., Spitzer L., Szilárd V., Viola R.