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Question Number 131430 by mathlove last updated on 04/Feb/21

$$\sqrt[\frac{1}{x}]{(\sqrt[\frac{1}{x}]{{x{}^x)}^x}}=4findx=?$$

Commented by math35 last updated on 04/Feb/21

$$\sqrt[\frac{1}{x}]{(\sqrt[\frac{1}{x}]{x^{x^2}}}=4$$$$note:\sqrt[\frac{1}{n}]{y}=y^{\frac{1}{\left(\frac{1}{n}\right)}}=y^n$$$$\to \sqrt[\frac{1}{x}]{{\left(x^{x^2}\right)}^x}=4$$$${\left(x^{x^3}\right)}^x=4$$$$x^{x^4}=4$$$$takinglnofbothsides$$$$x^{4}lnx=ln4$$$$e^{{lnx}^4}.lnx=ln4$$$$e^{4lnx}.4lnx=4ln4$$$$e^{4lnx}.4lnx=e^{ln4}.ln4$$$$takinglambertWfunctionofbothsides$$$$W[e^{4lnx}.4lnx]=W[e^{ln4}.ln4]$$$$4lnx=ln4$$$$lnx=\frac{1}{4}ln{2}^2=\frac{2}{4}ln2$$$$lnx=\frac{1}{2}ln2$$$$lnx=ln\sqrt{2}$$$$x=\sqrt{2}$$$$$$$$$$

Commented by mathlove last updated on 05/Feb/21

$$whatistheWfunction?$$

Answered by talminator2856791 last updated on 04/Feb/21

$$x=4$$

Commented by mathlove last updated on 04/Feb/21

$$pleassolve$$

Answered by mr W last updated on 05/Feb/21

$$iinterpretthequestionas:$$$$\sqrt[\frac{1}{x}]{{\left(\sqrt[\frac{1}{x}]{x^x}\right)}^x}=4$$$$\iff \sqrt[\frac{1}{x}]{{\left({\left(x^x\right)}^x\right)}^x}=4$$$$\iff {\left({\left({\left(x^x\right)}^x\right)}^x\right)}^x=4$$$$\iff {\left({\left(x^{x^2}\right)}^x\right)}^x=4$$$$\iff {\left(x^{x^3}\right)}^x=4$$$$\iff x^{x^4}=4$$$$\Rightarrow x^4=4$$$$\Rightarrow x=\sqrt[4]{4}=\sqrt{2}>0$$

Commented by mathlove last updated on 05/Feb/21

$$thanksalotsir$$

Commented by mr W last updated on 05/Feb/21

$$x^{x^{x^{{\text{iddots}}^{x^a}}}}=a$$$$hasthesamesolutionas{x}^a=a,$$$$i.e.x=\sqrt[a]{a}$$

Commented by Tawa11 last updated on 23/Jul/21

$$\mathrm{great}$$

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