By what should the expression

$$
\sqrt[3]{5 \sqrt{3}-3 \sqrt{7}}
$$

be multiplied so that the value of the product is 2?
I. solution. It is directly clear that $\frac{2}{\sqrt[3]{5 \sqrt{3}-3 \sqrt{7}}}$ is the sought number.

$$
\frac{2}{\sqrt[3]{5 \sqrt{3}-3 \sqrt{7}}}=\frac{2 \sqrt[3]{5 \sqrt{3}+3 \sqrt{7}}}{\sqrt[3]{5^{2} \cdot 3-3^{2} \cdot 7}}=\frac{2 \sqrt[3]{5 \sqrt{3}+3 \sqrt{7}}}{\sqrt[3]{12}}=\sqrt[3]{\frac{2}{3}(5 \sqrt{3}+3 \sqrt{7})}
$$

(György Ödön Singer, Budapest.)

II. solution. Since

$$
\sqrt[3]{5 \sqrt{3}-3 \sqrt{7}} \cdot \sqrt[3]{5 \sqrt{3}+3 \sqrt{7}}=\sqrt[3]{5^{2} \cdot 3-3^{2} \cdot 7}=\sqrt[3]{12}=\sqrt[3]{\frac{3}{2}} \cdot 8=2 \sqrt{\frac{3}{2}}
$$

therefore

$$
\sqrt[3]{\frac{2}{3}(5 \sqrt{3}+3 \sqrt{7})}
$$

is the sought expression.

(Solymári mathematical circle.)