Menu Close

Question-193626




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

Leave a Reply

Your email address will not be published. Required fields are marked *