Question Number 103686 by Jamshidbek2311 last updated on 16/Jul/20
$${x}^{{x}^{\mathrm{6}} } =\sqrt{\mathrm{2}}\:^{\sqrt{\mathrm{2}}} \:{x}=? \\ $$
Answered by Dwaipayan Shikari last updated on 16/Jul/20
$${x}^{\mathrm{6}} {logx}=\sqrt{\mathrm{2}}.\frac{\mathrm{1}}{\mathrm{2}}{log}\mathrm{2} \\ $$$$\mathrm{6}{logxe}^{\mathrm{6}{logx}} =\mathrm{3}\sqrt{\mathrm{2}}{log}\mathrm{2} \\ $$$$\mathrm{6}{logx}={W}_{\mathrm{0}} \left(\mathrm{3}\sqrt{\mathrm{2}}{log}\mathrm{2}\right) \\ $$$${x}={e}^{\frac{{W}_{\mathrm{0}} \left(\mathrm{3}\sqrt{\mathrm{2}}{log}\mathrm{2}\right)}{\mathrm{6}}} =\mathrm{1}.\mathrm{189}.... \\ $$$$ \\ $$