x-x-6-729-Find-x-

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

\sqrt{x^{x^{6}}} = 729, Find x

Answered by mrW1 last updated on 28/Jun/17

\sqrt{x^{x^{6}}} = 729

x^{x^{6}} = 729^{2}

t = x^{6}

x = t^{\frac{1}{6}}

\Rightarrow {(t^{\frac{1}{6}})}^{t} = 729^{2}

t^{\frac{t}{6}} = 729^{2} = a

t = a^{\frac{6}{t}} = e^{\frac{6}{t} \ln a}

\frac{1}{t} e^{\frac{6}{t} \ln a} = 1

\frac{6 \ln a}{t} e^{\frac{6}{t} \ln a} = 6 \ln a

\Rightarrow \frac{6 \ln a}{t} = W (6 \ln a)

\Rightarrow t = x^{6} = \frac{6 \ln a}{W (6 \ln a)} = \frac{6 \ln 729^{2}}{W (6 \ln 729^{2})} = \frac{12 \ln 729}{W (12 \ln 729)}

\Rightarrow x =^{6} \sqrt{\frac{12 \ln 729}{W (12 \ln 729)}}

\approx^{6} \sqrt{\frac{79.1}{W (79.1)}}

\approx^{6} \sqrt{\frac{79.1}{3.20576}}

\approx 1.70624

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

Wow, God bless you sir . I really appreciate .