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Question Number 16909 by tawa tawa last updated on 28/Jun/17

(√x^x^6 ) = 729 , Find x

$$\sqrt{\mathrm{x}^{\mathrm{x}^{\mathrm{6}} } }\:=\:\mathrm{729}\:,\:\:\:\:\mathrm{Find}\:\mathrm{x} \\ $$

Answered by mrW1 last updated on 28/Jun/17

(√x^x^6 ) = 729 x^x^6 =729^2 t=x^6 x=t^(1/6) ⇒(t^(1/6) )^t =729^2 t^(t/6) =729^2 =a t=a^(6/t) =e^((6/t)ln a) (1/t)e^((6/t)ln a) =1 ((6ln a)/t)e^((6/t)ln a) =6ln a ⇒((6ln a)/t)=W(6ln a) ⇒t=x^6 =((6ln a)/(W(6ln a)))=((6ln 729^2 )/(W(6ln 729^2 )))=((12ln 729)/(W(12ln 729))) ⇒x=^6 (√((12 ln 729)/(W(12 ln 729)))) ≈^6 (√((79.1)/(W(79.1)))) ≈^6 (√((79.1)/(3.20576))) ≈1.70624

$$\sqrt{\mathrm{x}^{\mathrm{x}^{\mathrm{6}} } }\:=\:\mathrm{729} \\ $$$$\mathrm{x}^{\mathrm{x}^{\mathrm{6}} } =\mathrm{729}^{\mathrm{2}} \\ $$$$\mathrm{t}=\mathrm{x}^{\mathrm{6}} \\ $$$$\mathrm{x}=\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{6}}} \\ $$$$\Rightarrow\left(\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{6}}} \right)^{\mathrm{t}} =\mathrm{729}^{\mathrm{2}} \\ $$$$\mathrm{t}^{\frac{\mathrm{t}}{\mathrm{6}}} =\mathrm{729}^{\mathrm{2}} =\mathrm{a} \\ $$$$\mathrm{t}=\mathrm{a}^{\frac{\mathrm{6}}{\mathrm{t}}} =\mathrm{e}^{\frac{\mathrm{6}}{\mathrm{t}}\mathrm{ln}\:\mathrm{a}} \\ $$$$\frac{\mathrm{1}}{\mathrm{t}}\mathrm{e}^{\frac{\mathrm{6}}{\mathrm{t}}\mathrm{ln}\:\mathrm{a}} =\mathrm{1} \\ $$$$\frac{\mathrm{6ln}\:\mathrm{a}}{\mathrm{t}}\mathrm{e}^{\frac{\mathrm{6}}{\mathrm{t}}\mathrm{ln}\:\mathrm{a}} =\mathrm{6ln}\:\mathrm{a} \\ $$$$\Rightarrow\frac{\mathrm{6ln}\:\mathrm{a}}{\mathrm{t}}=\mathrm{W}\left(\mathrm{6ln}\:\mathrm{a}\right) \\ $$$$\Rightarrow\mathrm{t}=\mathrm{x}^{\mathrm{6}} =\frac{\mathrm{6ln}\:\mathrm{a}}{\mathrm{W}\left(\mathrm{6ln}\:\mathrm{a}\right)}=\frac{\mathrm{6ln}\:\mathrm{729}^{\mathrm{2}} }{\mathrm{W}\left(\mathrm{6ln}\:\mathrm{729}^{\mathrm{2}} \right)}=\frac{\mathrm{12ln}\:\mathrm{729}}{\mathrm{W}\left(\mathrm{12ln}\:\mathrm{729}\right)} \\ $$$$\Rightarrow\mathrm{x}=\:^{\mathrm{6}} \sqrt{\frac{\mathrm{12}\:\mathrm{ln}\:\mathrm{729}}{\mathrm{W}\left(\mathrm{12}\:\mathrm{ln}\:\mathrm{729}\right)}} \\ $$$$\approx\:^{\mathrm{6}} \sqrt{\frac{\mathrm{79}.\mathrm{1}}{\mathrm{W}\left(\mathrm{79}.\mathrm{1}\right)}} \\ $$$$\approx\:^{\mathrm{6}} \sqrt{\frac{\mathrm{79}.\mathrm{1}}{\mathrm{3}.\mathrm{20576}}} \\ $$$$\approx\mathrm{1}.\mathrm{70624} \\ $$

Commented by tawa tawa last updated on 28/Jun/17

Wow, God bless you sir. I really appreciate.

$$\mathrm{Wow},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{really}\:\mathrm{appreciate}.\: \\ $$

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