solve-x-3-1-1-x-27-

Question Number 156709 by aliyn last updated on 14/Oct/21

s o l v e \sqrt[\frac{1}{x}]{x^{3}} = 27

Answered by mr W last updated on 14/Oct/21

\sqrt[\frac{1}{x}]{x^{3}} = {(x^{3})}^{x} = {(x^{x})}^{3} = 27 = 3^{3}

\Rightarrow x^{x} = 3

\Rightarrow x = 3^{\frac{1}{x}} = e^{\frac{\ln 3}{x}}

\Rightarrow \ln 3 = \frac{\ln 3}{x} e^{\frac{\ln 3}{x}}

\Rightarrow \frac{\ln 3}{x} = W (\ln 3)

\Rightarrow x = \frac{\ln 3}{W (\ln 3)} \approx \frac{\ln 3}{0.60182928} = 1.825455

Commented by tabata last updated on 14/Oct/21

t h a n k y o u s i r . b u t c a n y o u g i v e m e f o r m u l l a

h o w c a n c a l c u l a t e \sqrt[\frac{1}{a n y v e r i a b l e}]{a n y v e r i a b l e}

Commented by mr W last updated on 14/Oct/21

i t h i n k y o u k n o w \sqrt[3]{2} = 2^{\frac{1}{3}} . g e n e r a l l y

\sqrt[b]{a} = a^{\frac{1}{b}}

w h e n y o u r e p l a c e b w i t h \frac{1}{c}, y o u h a v e

\sqrt[\frac{1}{c}]{a} = a^{\frac{1}{(\frac{1}{c})}} = a^{c}

Commented by tabata last updated on 14/Oct/21

t h a n k y o u s i r

Commented by mr W last updated on 15/Oct/21

o u i!

Commented by Sky_b last updated on 15/Oct/21

l a f o n t i o n W d e l a m b e r t ? ?

Commented by Ruuudiy last updated on 16/Oct/21

Ans

I think you know \sqrt[x]{y} = y^{\frac{1}{x}}

But if you replace x with \frac{1}{z} \dots

\sqrt[\frac{1}{z}]{y} = y^{\frac{1}{\frac{1}{z}}} = y^{z}

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