Problem 4. Simplify the expression

$$
M=\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}
$$
Solution. Using a calculator, we find the approximate value of the expression $M$. To simplify the calculations, we transform the expression

$$
K=4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}
$$

as follows:

$$
\begin{gathered}
K=4-\sqrt[4]{405}+\sqrt[4]{400}-\sqrt[4]{125}=4+\sqrt[4]{400}- \\
-(\sqrt[4]{405}+\sqrt[4]{125})
\end{gathered}
$$

On the calculator, we get:

$$
K \approx 0.6423887 \text { and } M \approx 2.4953477 .
$$

Since the expression $M$ contains $\sqrt[4]{5}, \sqrt[4]{25}=(\sqrt[4]{5})^{2}$, and $\sqrt[4]{125}=(\sqrt[4]{5})^{3}$, the transformed expression will also contain $\sqrt[4]{5}$. But $\sqrt[4]{5} \approx 1.4953487$ (result obtained on the calculator).

Thus, as a result of the mathematical experiment, we hypothesize that $M=\sqrt[4]{5}+1$.

Let's try to prove that

$$
\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}=\sqrt[4]{5}+1
$$

or

$$
\frac{2}{\sqrt[4]{5}+1}=\sqrt{4-3 \sqrt[4]{5+2 \sqrt[4]{25}-\sqrt[4]{125}}}
$$

If equality (3) is true, then after squaring both sides, we get:

$$
\frac{4}{\sqrt[4]{25}+2 \sqrt[4]{5}+1}=4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}
$$

or

$$
\begin{gathered}
4=(\sqrt[4]{25}+2 \sqrt[4]{5}+1) \times \\
\times(4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}) \\
4=(\sqrt[4]{25}-3 \sqrt[4]{125}+2 \sqrt[4]{625}-\sqrt[4]{25 \cdot 125}+ \\
+8 \sqrt[4]{5}-6 \sqrt[4]{25}+4 \sqrt[4]{125}-2 \sqrt[4]{625}+4- \\
-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}
\end{gathered}
$$

The last equality is true. Thus, $M=\sqrt[4]{5}+1$.