3 Task 1. For which natural $n(n \geqslant 2)$ is the equality

$$
-\sqrt[2]{17 \sqrt{5}+38}+\sqrt[2]{17 \sqrt{5}-38}=\sqrt{20} ?
$$
Solution. Using a calculator, we find:

$$
\begin{gathered}
17 \sqrt{5}+38 \approx 38.013154+38 \approx 76.013154>1 \\
17 \sqrt{5}-38 \approx 0.0131541$, and the function $\varphi(n)=\sqrt[n]{17 \sqrt{5}-38}$ is increasing, and $\varphi(n)<1$. To discover some properties of the function $\psi(n)=f(n)+\varphi(n)$, we will perform a mathematical experiment (construct a table of values of the function $\psi(n)$):

| $n$ | $f(n)$ | $\varphi(n)$ | $\varphi(n)$ |
| :---: | :---: | :---: | :---: |
| 2 | 8.7185522 | 0.1146908 | 8.8332422 |
| 3 | 4.2358226 | 0.2360586 | 4.4718813 |
| 4 | 2.952194 | 0.3386604 | 3.2913798 |
| 5 | 2.3777310 | 0.4205441 | 2.8002750 |

Now it is clear that equation (1) has a unique solution $n=3$.