Problem 2. Solve the system $\left\{\begin{array}{l}\sqrt[4]{x \sqrt{x}}-\sqrt[4]{y \sqrt{y}}=7 \\ \sqrt[14]{x \sqrt{x \sqrt{x}}}+\sqrt[14]{y \sqrt{y \sqrt{y}}}=3\end{array}\right.$.

Mathematical Gazette $n$ r. 9/2013
Solution. Clearly, $x, y \geq 0$. The system can be rewritten as

$$
\left\{\begin{array}{l}
\sqrt[8]{x^{3}}-\sqrt[8]{y^{3}}=7 \\
\sqrt[8]{x}+\sqrt[8]{y}=3
\end{array}\right.
$$

Let $\sqrt[8]{x}=a, \sqrt[8]{y}=b$, where $a, b \geq 0$, then we have $a+b=3$ and $a^{3}-b^{3}=7$, which leads to the equation

$$
a^{3}-(3-a)^{3}=7 \Leftrightarrow 2 a^{3}-9 a^{2}+27 a-34=0 \Leftrightarrow(a-2)\left(2 a^{2}-5 a+17\right)=0
$$

The only real solution to the equation is $a=2$, for which we get $b=1, x=256$ and $y=1$.