11.1. Does there exist a natural number $n$, greater than 1, such that the value of the expression $\sqrt{n \sqrt{n \sqrt{n}}}$ is a natural number?
Answer: Yes, it exists.

Solution. For example, $n=2^{8}=256$.

Indeed, $\sqrt{n \sqrt{n \sqrt{n}}}=\sqrt{n \sqrt{n \cdot n^{\frac{1}{2}}}}=\sqrt{n \sqrt{n^{\frac{3}{2}}}}=\sqrt{n \cdot n^{\frac{3}{4}}}=\sqrt{n^{\frac{7}{4}}}=n^{\frac{7}{8}}$. Then, for $n=2^{8}$, the value of this expression is $\left(2^{8}\right)^{\frac{7}{8}}=2^{7}=128$.

Other examples are possible: any numbers that are the eighth power of any natural number other than 1.

## Grading Criteria

+ A complete and justified solution is provided

$\pm$ A correct example is provided, but the calculation showing that it fits is not shown

- Only the answer ("yes" or "no") is provided
- The problem is not solved or is solved incorrectly