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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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