Question Number 132260 by Salman_Abir last updated on 12/Feb/21
Answered by Olaf last updated on 13/Feb/21
![(x(√x))^x = x^(x(√x)) (x^(3/2) )^x = x^x^(3/2) xlnx^(3/2) = x^(3/2) lnx xlnx^(3/2) = x^(3/2) lnx (3/2)xlnx = x^(3/2) lnx xlnx[(3/2)−(√x)] = 0 (necessarilly x≠0) x = 1 or x = (9/4)](https://www.tinkutara.com/question/Q132291.png)
$$\left({x}\sqrt{{x}}\right)^{{x}} \:=\:{x}^{{x}\sqrt{{x}}} \\ $$$$\left({x}^{\mathrm{3}/\mathrm{2}} \right)^{{x}} \:=\:{x}^{{x}^{\mathrm{3}/\mathrm{2}} } \\ $$$${x}\mathrm{ln}{x}^{\mathrm{3}/\mathrm{2}} \:=\:{x}^{\mathrm{3}/\mathrm{2}} \mathrm{ln}{x} \\ $$$${x}\mathrm{ln}{x}^{\mathrm{3}/\mathrm{2}} \:=\:{x}^{\mathrm{3}/\mathrm{2}} \mathrm{ln}{x} \\ $$$$\frac{\mathrm{3}}{\mathrm{2}}{x}\mathrm{ln}{x}\:=\:{x}^{\mathrm{3}/\mathrm{2}} \mathrm{ln}{x} \\ $$$${x}\mathrm{ln}{x}\left[\frac{\mathrm{3}}{\mathrm{2}}−\sqrt{{x}}\right]\:=\:\mathrm{0} \\ $$$$\left(\mathrm{necessarilly}\:{x}\neq\mathrm{0}\right) \\ $$$${x}\:=\:\mathrm{1}\:\mathrm{or}\:{x}\:=\:\frac{\mathrm{9}}{\mathrm{4}} \\ $$
Commented by otchereabdullai@gmail.com last updated on 13/Feb/21
$$\mathrm{nice}\:\mathrm{one}\:\mathrm{sir}! \\ $$