Question Number 120545 by pooooop last updated on 01/Nov/20
$$\left(\sqrt{\boldsymbol{{x}}}\right)^{\frac{\boldsymbol{{x}}}{\:\sqrt{\boldsymbol{{x}}}}} \:=\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\boldsymbol{{x}}=? \\ $$
Commented by pooooop last updated on 01/Nov/20
$$ \\ $$$$\:\:\:\left(\sqrt{{x}}\right)^{\frac{{x}\sqrt{{x}}}{{x}}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$
Commented by pooooop last updated on 01/Nov/20
$$\:\:\:\:\:\sqrt{{x}}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:=>\:\:\:{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$
Answered by Jamshidbek2311 last updated on 01/Nov/20
$$\sqrt{{x}}={a}\geqslant\mathrm{0} \\ $$$${a}^{\frac{{a}^{\mathrm{2}} }{{a}}} =\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\Rightarrow\:{a}^{{a}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:\Rightarrow\:{a}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\sqrt{{x}}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\Rightarrow\:{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$ \\ $$