Prove $4-\sqrt{2}-\sqrt[3]{3}-\sqrt[5]{5} \gt 0$ Is it possible to know if $4-\sqrt{2}-\sqrt[3]{3}-\sqrt[5]{5} \gt 0$ without using decimal numbers?

It is not hard to verify following inequalities (just power both sides and it should result into simple inequalities in natural numbers only):
\begin{align}
\frac{4}{3} &< \sqrt{2} < \frac{5}{3}\\
\frac{4}{3} &< \sqrt[3]{3} < \frac{5}{3}\\
\frac{4}{3} &< \sqrt[5]{5} < \frac{5}{3}\\
\end{align}
Summing these up will give you
$$
4 < \sqrt{2}+\sqrt[3]{3}+\sqrt[5]{5} < 5\\
$$
