Problem 4. Solve in the set of real numbers

$$
\sqrt[3]{x \sqrt{x}}-\sqrt[7]{x \sqrt{x \sqrt{x}}}=6 \sqrt[12]{x \sqrt{x}}
$$

Mathematical Gazette
Solution:

Condition for the existence of the radical: $x \geq 0$

![](https://cdn.mathpix.com/cropped/2024_06_07_5409dda2413c1d8c519ag-3.jpg?height=93&width=1607&top_left_y=1054&top_left_x=240)

![](https://cdn.mathpix.com/cropped/2024_06_07_5409dda2413c1d8c519ag-3.jpg?height=65&width=1602&top_left_y=1174&top_left_x=240)

![](https://cdn.mathpix.com/cropped/2024_06_07_5409dda2413c1d8c519ag-3.jpg?height=68&width=1602&top_left_y=1283&top_left_x=240)

![](https://cdn.mathpix.com/cropped/2024_06_07_5409dda2413c1d8c519ag-3.jpg?height=71&width=1631&top_left_y=1393&top_left_x=239)

Or $t^{2}+2 t+3=0 \Rightarrow \Delta_{t}=-8<0 \Rightarrow$ the equation has no real solutions. ..........................5p

![](https://cdn.mathpix.com/cropped/2024_06_07_5409dda2413c1d8c519ag-3.jpg?height=62&width=1631&top_left_y=1606&top_left_x=239)