Is it possible to know if $4-\sqrt{2}-\sqrt[3]{3}-\sqrt[5]{5} \gt 0$ without using decimal numbers?
2026-04-04 00:15:22.1775261722
Prove $4-\sqrt{2}-\sqrt[3]{3}-\sqrt[5]{5} \gt 0$
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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 $$ 4 < \sqrt{2}+\sqrt[3]{3}+\sqrt[5]{5} < 5\\ $$