Question

Solve the equation $5^x\sqrt[x]{x{8}^{x-1}}=500$
find the rational root .

Answer 1

Given equation $5^x\sqrt[x]{x{8}^{x-1}}=500$
$\Rightarrow 5^x{.8}^{\left(\frac{x-1}{x}\right)}=5^3.2^2$
$\Rightarrow 5^x.2^{\frac{3x-3}{x}}=5^3.2^2$
$5^{x-3}.2^{\left(\frac{x-3}{x}\right)}=1$
${\left({5.2}^{1/x}\right)}^{(x-3)}=1$
$\Rightarrow (x-3)\log \left({5.2}^{1/x}\right)=0$
$\Rightarrow x-3=0,\log \left({5.2}^{1/x}\right)=0$
$\therefore x_1=3,{5.2}^{1/x}=1$
$\Rightarrow 2^{1/x}=\left(1/5\right)$
$\therefore x_2=-lo{g}_{5}2$
Hence roots of the original equation are $x_1=3$ and $x_2=-lo{g}_{5}2$