solve-this-pls-x-x-1-2-1-2-

Question Number 106076 by Skabetix last updated on 02/Aug/20

s o l v e t h i s p l s

x^{x^{\frac{1}{2}}} = \frac{1}{2}

Answered by Dwaipayan Shikari last updated on 02/Aug/20

A n o t h e r m e t h o d

x^{x^{\frac{1}{2}}} = \frac{1}{2} (l o o k t h a t \frac{1}{2} = x^{x^{\frac{1}{2}}}

x^{x^{x^{x^{\frac{1}{2}}}}} = \frac{1}{2}

l i k e t h i s m a n n e r

x^{x^{x^{x^{x^{x^{x}}}}}} = \frac{1}{2}

s o x^{\frac{1}{2}} = \frac{1}{2}

x = \frac{1}{4}

I t i s g e n e r a l f o r

X^{X^{N}} = N

X = \sqrt[N]{N}

Commented by mr W last updated on 02/Aug/20

t h e r e a r e t w o r o o t s :

x = e^{2 W (- \frac{\ln 2}{2})} = {\begin{cases} \frac{1}{4} \\ \frac{1}{16} \end{cases}

Answered by Dwaipayan Shikari last updated on 02/Aug/20

x^{\frac{1}{2}} l o g x = l o g (\frac{1}{2})

l o g x . e^{\frac{1}{2} l o g (x)} = l o g (\frac{1}{2})

\frac{1}{2} l o g (x) e^{\frac{1}{2} l o g (x)} = \frac{1}{2} l o g (\frac{1}{2})

\frac{1}{2} l o g (x) = W_{0} (\frac{1}{2} l o g (\frac{1}{2}))

x = e^{2 W_{0} (- l o g (\sqrt{2})} = \frac{1}{4}, \frac{1}{16}