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Find-the-value-of-x-if-x-x-6-729-




Question Number 5385 by sanusihammed last updated on 12/May/16
Find the value of x if     (√x^x^6  ) =  729
$${Find}\:{the}\:{value}\:{of}\:{x}\:{if}\: \\ $$$$ \\ $$$$\sqrt{{x}^{{x}^{\mathrm{6}} } }\:=\:\:\mathrm{729} \\ $$
Answered by prakash jain last updated on 12/May/16
(√x^x^6  ) =729=3^6   x^x^6  =3^(12)   x^x^6  =(3^2 )^6   x^x =9  If the question was cube root on LHS  (x^x^6  )^(1/3) =729  x^x^6  =3^(18)   x^x =3^3 ⇒x=3
$$\sqrt{{x}^{{x}^{\mathrm{6}} } }\:=\mathrm{729}=\mathrm{3}^{\mathrm{6}} \\ $$$${x}^{{x}^{\mathrm{6}} } =\mathrm{3}^{\mathrm{12}} \\ $$$${x}^{{x}^{\mathrm{6}} } =\left(\mathrm{3}^{\mathrm{2}} \right)^{\mathrm{6}} \\ $$$${x}^{{x}} =\mathrm{9} \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{question}\:\mathrm{was}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{on}\:\mathrm{LHS} \\ $$$$\sqrt[{\mathrm{3}}]{{x}^{{x}^{\mathrm{6}} } }=\mathrm{729} \\ $$$${x}^{{x}^{\mathrm{6}} } =\mathrm{3}^{\mathrm{18}} \\ $$$${x}^{{x}} =\mathrm{3}^{\mathrm{3}} \Rightarrow{x}=\mathrm{3} \\ $$
Commented by Rasheed Soomro last updated on 16/May/16
x^x^6  =^(?)  (x^x )^6   Or x^x^6  =x^((x^6 ))  ?  x^x^6  =(3^2 )^6 ⇒^(?) x^x =9
$$\mathrm{x}^{\mathrm{x}^{\mathrm{6}} } \overset{?} {=}\:\left(\mathrm{x}^{\mathrm{x}} \right)^{\mathrm{6}} \\ $$$$\mathrm{Or}\:\mathrm{x}^{\mathrm{x}^{\mathrm{6}} } =\mathrm{x}^{\left(\mathrm{x}^{\mathrm{6}} \right)} \:? \\ $$$${x}^{{x}^{\mathrm{6}} } =\left(\mathrm{3}^{\mathrm{2}} \right)^{\mathrm{6}} \overset{?} {\Rightarrow}{x}^{{x}} =\mathrm{9} \\ $$$$ \\ $$
Answered by Rasheed Soomro last updated on 15/May/16
(√x^x^6  ) =  729⇒x^((1/2)x^6 ) =729  −−−−−−−−−−−−  To write 729 in the base of x  x^■ =729  ■log_3  x=log_3 729=6  ■=(6/(log_3  x))  729=x^(6/(log_3  x))   −−−−−−−−−  x^((1/2)x^6 ) = x^(6/(log_3  x)) ⇒(1/2)x^6 =(6/(log_3  x))  (1/2)x^6  log_3  x=6  x^6 log_3 x=12  x^6 =((3×4)/(log_3 x))=((log_3 27 ×4)/(log_3 x))=((log_3 27^4 )/(log_3 x))=log_x 27^4   x^6 =log_x 27^4   Continue
$$\sqrt{{x}^{{x}^{\mathrm{6}} } }\:=\:\:\mathrm{729}\Rightarrow{x}^{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{6}} } =\mathrm{729} \\ $$$$−−−−−−−−−−−− \\ $$$$\mathrm{To}\:\mathrm{write}\:\mathrm{729}\:\mathrm{in}\:\mathrm{the}\:\mathrm{base}\:\mathrm{of}\:\mathrm{x} \\ $$$${x}^{\blacksquare} =\mathrm{729} \\ $$$$\blacksquare\mathrm{log}_{\mathrm{3}} \:\mathrm{x}=\mathrm{log}_{\mathrm{3}} \mathrm{729}=\mathrm{6} \\ $$$$\blacksquare=\frac{\mathrm{6}}{\mathrm{log}_{\mathrm{3}} \:\mathrm{x}} \\ $$$$\mathrm{729}=\mathrm{x}^{\frac{\mathrm{6}}{\mathrm{log}_{\mathrm{3}} \:\mathrm{x}}} \\ $$$$−−−−−−−−− \\ $$$${x}^{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{6}} } =\:\mathrm{x}^{\frac{\mathrm{6}}{\mathrm{log}_{\mathrm{3}} \:\mathrm{x}}} \Rightarrow\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{6}} =\frac{\mathrm{6}}{\mathrm{log}_{\mathrm{3}} \:\mathrm{x}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{6}} \:\mathrm{log}_{\mathrm{3}} \:\mathrm{x}=\mathrm{6} \\ $$$${x}^{\mathrm{6}} \mathrm{log}_{\mathrm{3}} \mathrm{x}=\mathrm{12} \\ $$$${x}^{\mathrm{6}} =\frac{\mathrm{3}×\mathrm{4}}{\mathrm{log}_{\mathrm{3}} \mathrm{x}}=\frac{\mathrm{log}_{\mathrm{3}} \mathrm{27}\:×\mathrm{4}}{\mathrm{log}_{\mathrm{3}} \mathrm{x}}=\frac{\mathrm{log}_{\mathrm{3}} \mathrm{27}^{\mathrm{4}} }{\mathrm{log}_{\mathrm{3}} \mathrm{x}}=\mathrm{log}_{\mathrm{x}} \mathrm{27}^{\mathrm{4}} \\ $$$$\mathrm{x}^{\mathrm{6}} =\mathrm{log}_{\mathrm{x}} \mathrm{27}^{\mathrm{4}} \\ $$$$\mathrm{Continue} \\ $$

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