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Question Number 120545 by pooooop last updated on 01/Nov/20

((√x))^(x/( (√x)))  =  (1/( (√2)))     x=?

$$\left(\sqrt{\boldsymbol{{x}}}\right)^{\frac{\boldsymbol{{x}}}{\:\sqrt{\boldsymbol{{x}}}}} \:=\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\boldsymbol{{x}}=? \\ $$

Commented by pooooop last updated on 01/Nov/20

     ((√x))^((x(√x))/x) =((1/2))^(1/2)

$$ \\ $$$$\:\:\:\left(\sqrt{{x}}\right)^{\frac{{x}\sqrt{{x}}}{{x}}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$

Commented by pooooop last updated on 01/Nov/20

     (√x)=(1/2)    =>   x=(1/4)

$$\:\:\:\:\:\sqrt{{x}}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:=>\:\:\:{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Answered by Jamshidbek2311 last updated on 01/Nov/20

(√x)=a≥0  a^(a^2 /a) =(1/(√2))   ⇒ a^a =((1/2))^(1/2)  ⇒ a=(1/2)  (√x)=(1/2)   ⇒ x=(1/4)

$$\sqrt{{x}}={a}\geqslant\mathrm{0} \\ $$$${a}^{\frac{{a}^{\mathrm{2}} }{{a}}} =\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:\:\:\Rightarrow\:{a}^{{a}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:\Rightarrow\:{a}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\sqrt{{x}}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\Rightarrow\:{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$ \\ $$

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