1. Find all natural numbers $n$ such that the value of the expression $\sqrt{n \sqrt{n \sqrt{n}}}$ is a natural number less than 2217.

Answer: $n=3^{8}, n=2^{8}$.
Solution. Transform the expression $\sqrt{n \sqrt{n \sqrt{n}}}=\left(n\left(n(n)^{\frac{1}{2}}\right)^{\frac{1}{2}}\right)^{\frac{1}{2}}=$

$\left(n\left(n^{\frac{3}{2}}\right)^{\frac{1}{2}}\right)^{\frac{1}{2}}=\left(n(n)^{\frac{3}{4}}\right)^{\frac{1}{2}}=\left(n^{\frac{7}{4}}\right)^{\frac{1}{2}}=n^{\frac{7}{8}}$.

This means the number must be the eighth power of some natural number. For $n=4^{8}=2^{16}$, the check gives:

$\sqrt{2^{16} \sqrt{2^{16} \sqrt{2^{16}}}}=\sqrt{2^{16} \sqrt{2^{24}}}=\sqrt{2^{28}}=2^{14}=4096>2217$.

It remains to check only $n=3^{8}, n=2^{8}$.

$\sqrt{3^{8} \sqrt{3^{8} \sqrt{3^{8}}}}=\sqrt{3^{8} \sqrt{3^{12}}}=\sqrt{3^{14}}=3^{7}=2187<2217$.

$\sqrt{2^{8} \sqrt{2^{8} \sqrt{2^{8}}}}=\sqrt{2^{8} \sqrt{2^{12}}}=\sqrt{2^{14}}=2^{7}=128<2217$.

Criteria. If the solution is incorrect - 0 points.

Only the correct answer - 1 point.

If both correct examples are provided and it is verified that they fit, but it is not proven that there are no other answers - 3 points.

Only the correct estimate is provided - 3 points.

If the solution is correct - 7 points.