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Question Number 193626 by SaRahAli last updated on 17/Jun/23

Answered by MM42 last updated on 17/Jun/23

x=(√2) ⇒((√2))^(((√2))^8 ) =((√2))^(16) =256 & ((√2))^(((√2))^2 ) =((√2))^2 =2 ⇒and=258

$${x}=\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{8}} } =\left(\sqrt{\mathrm{2}}\right)^{\mathrm{16}} =\mathrm{256}\:\:\&\:\:\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} } =\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} =\mathrm{2} \\ $$$$\Rightarrow{and}=\mathrm{258} \\ $$$$ \\ $$

Commented by aba last updated on 17/Jun/23

((√2))^(16) =2^8 =256

$$\left(\sqrt{\mathrm{2}}\right)^{\mathrm{16}} =\mathrm{2}^{\mathrm{8}} =\mathrm{256} \\ $$

Commented by MM42 last updated on 17/Jun/23

yes.

$${yes}. \\ $$

Answered by aba last updated on 17/Jun/23

x^x^4 =4 ⇒ x^4 lnx=2ln2 ⇒ 4lnx.e^(4lnx) =8ln2 ⇒ 4lnx=W(8ln2)=W(2ln2.e^(2ln2) )=2ln2 ⇒ lnx=(1/2)ln2 ⇒ x=(√2) ((√2))^(((√2))^8 ) +((√2))^(((√2))^2 ) =((√2))^(((√2))^(2×4) ) +((√2))^(((√2))^2 ) =((√2))^2^4 +((√2))^(((√2))^2 ) =((√2))^(2×8) +((√2))^2 =2^8 +2 =258 ✓

$$\mathrm{x}^{\mathrm{x}^{\mathrm{4}} } =\mathrm{4}\:\Rightarrow\:\mathrm{x}^{\mathrm{4}} \mathrm{lnx}=\mathrm{2ln2}\:\Rightarrow\:\mathrm{4lnx}.\mathrm{e}^{\mathrm{4lnx}} =\mathrm{8ln2} \\ $$$$\Rightarrow\:\mathrm{4lnx}=\mathrm{W}\left(\mathrm{8ln2}\right)=\mathrm{W}\left(\mathrm{2ln2}.\mathrm{e}^{\mathrm{2ln2}} \right)=\mathrm{2ln2} \\ $$$$\Rightarrow\:\mathrm{lnx}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln2}\:\Rightarrow\:\mathrm{x}=\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{8}} } +\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} } =\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}×\mathrm{4}} } +\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} } =\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}^{\mathrm{4}} } +\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}×\mathrm{8}} +\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}^{\mathrm{8}} +\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{258}\:\checkmark \\ $$$$ \\ $$

Answered by mr W last updated on 17/Jun/23

x^x^4 =4 (x^4 )^x^4 =4^4 ⇒x^4 =4 ⇒x^2 =2 x^x^8 =x^((x^4 )^2 ) =x^(16) =(x^4 )^4 =4^4 =256 x^x^2 =x^2 =2 ⇒x^x^8 +x^x^2 =256+2=258

$${x}^{{x}^{\mathrm{4}} } =\mathrm{4} \\ $$$$\left({x}^{\mathrm{4}} \right)^{{x}^{\mathrm{4}} } =\mathrm{4}^{\mathrm{4}} \\ $$$$\Rightarrow{x}^{\mathrm{4}} =\mathrm{4} \\ $$$$\Rightarrow{x}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}^{{x}^{\mathrm{8}} } ={x}^{\left({x}^{\mathrm{4}} \right)^{\mathrm{2}} } ={x}^{\mathrm{16}} =\left({x}^{\mathrm{4}} \right)^{\mathrm{4}} =\mathrm{4}^{\mathrm{4}} =\mathrm{256} \\ $$$${x}^{{x}^{\mathrm{2}} } ={x}^{\mathrm{2}} =\mathrm{2} \\ $$$$\Rightarrow{x}^{{x}^{\mathrm{8}} } +{x}^{{x}^{\mathrm{2}} } =\mathrm{256}+\mathrm{2}=\mathrm{258} \\ $$

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