Question Number 190480 by aba last updated on 03/Apr/23
$$\mathrm{Proof}\:\mathrm{that}\:\left(\sqrt{\mathrm{2}}\right)^{\sqrt{\mathrm{2}}} \in\mathbb{R}\backslash\mathrm{Q} \\ $$
Answered by mehdee42 last updated on 04/Apr/23
$${lem}:\:{if}\:\:{p}\notin{Q}\Rightarrow\sqrt{{p}}\notin{Q} \\ $$$${clim}:\:\mathrm{2}^{\sqrt{\mathrm{2}}} \notin{Q} \\ $$$${proof}\::\:{if}\:\:\mathrm{2}^{\sqrt{\mathrm{2}}} =\frac{{p}}{{q}}\in{Q}\Rightarrow{p}=\mathrm{2}^{\sqrt{\mathrm{2}}} {q}\:\: \\ $$$${p},{q}\in\mathbb{N}\Rightarrow\mathrm{2}^{\sqrt{\mathrm{2}}} \in\mathbb{N}#\:\Rightarrow\mathrm{2}^{\sqrt{\mathrm{2}}} \notin{Q}\: \\ $$$$\Rightarrow\sqrt{\left(\mathrm{2}^{\sqrt{\mathrm{2}}} \right)}=\left(\sqrt{\mathrm{2}}\right)^{\sqrt{\mathrm{2}}} \notin{Q} \\ $$$$ \\ $$