x-x-x-1-1-x-1-1-x-4-find-x-

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

\sqrt[\frac{1}{x}]{(\sqrt[\frac{1}{x}]{{x^{x})}^{x}}} = 4 f i n d x = ?

Commented by math35 last updated on 04/Feb/21

\sqrt[\frac{1}{x}]{(\sqrt[\frac{1}{x}]{x^{x^{2}}}} = 4

n o t e : \sqrt[\frac{1}{n}]{y} = y^{\frac{1}{(\frac{1}{n})}} = y^{n}

\to \sqrt[\frac{1}{x}]{{(x^{x^{2}})}^{x}} = 4

{(x^{x^{3}})}^{x} = 4

x^{x^{4}} = 4

t a k i n g l n o f b o t h s i d e s

x^{4} l n x = l n 4

e^{{l n x}^{4}} . l n x = l n 4

e^{4 l n x} . 4 l n x = 4 l n 4

e^{4 l n x} . 4 l n x = e^{l n 4} . l n 4

t a k i n g l a m b e r t W f u n c t i o n o f b o t h s i d e s

W [e^{4 l n x} . 4 l n x] = W [e^{l n 4} . l n 4]

4 l n x = l n 4

l n x = \frac{1}{4} l n 2^{2} = \frac{2}{4} l n 2

l n x = \frac{1}{2} l n 2

l n x = l n \sqrt{2}

x = \sqrt{2}

Commented by mathlove last updated on 05/Feb/21

w h a t i s t h e W f u n c t i o n ?

Answered by talminator2856791 last updated on 04/Feb/21

x = 4

Commented by mathlove last updated on 04/Feb/21

p l e a s s o l v e

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

i i n t e r p r e t t h e q u e s t i o n a s :

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

\Leftrightarrow \sqrt[\frac{1}{x}]{{({(x^{x})}^{x})}^{x}} = 4

\Leftrightarrow {({({(x^{x})}^{x})}^{x})}^{x} = 4

\Leftrightarrow {({(x^{x^{2}})}^{x})}^{x} = 4

\Leftrightarrow {(x^{x^{3}})}^{x} = 4

\Leftrightarrow 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

t h a n k s a l o t s i r

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

x^{x^{x^{\iddots^{x^{a}}}}} = a

h a s t h e s a m e s o l u t i o n a s x^{a} = a,

i . e . x = \sqrt[a]{a}

Commented by Tawa11 last updated on 23/Jul/21

great

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