6.055. $\sqrt{x \sqrt[5]{x}}+\sqrt[5]{x \sqrt{x}}=56$.

6.055. $\sqrt{x \sqrt[5]{x}}+\sqrt[5]{x \sqrt{x}}=56$.
## Solution.

Domain of definition: $x \geq 0$.

From the condition we have

$$
x^{\frac{6}{10}}-x^{\frac{3}{10}}=56 \Leftrightarrow\left(x^{\frac{3}{10}}\right)^{2}-x^{\frac{3}{10}}-56=0
$$

Let $x^{\frac{3}{10}}=y \geq 0$. The equation in terms of $y$ becomes $y^{2}-y-56=0$, from which $y_{1}=-7$ or $y_{2}=8 ; y_{1}=-7<0$ is not valid. Therefore, $x^{\frac{3}{10}}=8$. Hence, $x=8^{\frac{10}{3}}, x=\left(2^{3}\right)^{\frac{10}{3}}, x=2^{10}=1024$.

Answer: $x=1024$.