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Question Number 195002 by mathlove last updated on 22/Jul/23
x^(√x) =((√x))^x      x=?
$${x}^{\sqrt{{x}}} =\left(\sqrt{{x}}\right)^{{x}} \:\:\:\:\:{x}=? \\ $$
Answered by dimentri last updated on 22/Jul/23
   (x)^(√x)  = (x)^((1/2)x)     (1) (√x) = (1/2)x⇒x−2(√x) =0       (√x) ((√x)−2)=0⇒x=4     (2) x= 1    ∴ x ∈ {1, 4}
$$\:\:\:\left({x}\right)^{\sqrt{{x}}} \:=\:\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}{x}} \\ $$$$\:\:\left(\mathrm{1}\right)\:\sqrt{{x}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}{x}\Rightarrow{x}−\mathrm{2}\sqrt{{x}}\:=\mathrm{0} \\ $$$$\:\:\:\:\:\sqrt{{x}}\:\left(\sqrt{{x}}−\mathrm{2}\right)=\mathrm{0}\Rightarrow{x}=\mathrm{4} \\ $$$$\:\:\:\left(\mathrm{2}\right)\:{x}=\:\mathrm{1} \\ $$$$\:\:\therefore\:{x}\:\in\:\left\{\mathrm{1},\:\mathrm{4}\right\}\:\: \\ $$$$\:\: \\ $$
Commented by mathlove last updated on 22/Jul/23
thanks
$${thanks} \\ $$

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