Question

Solve the following equations:
$1+8x^{\frac{6}{5}}+9\sqrt[5]{5x^3}=0$.

Answer 1

$1+8x^{\frac{6}{5}}+9\sqrt[5]{5x^3}=01+8{\left(x^{\frac{3}{5}}\right)}^2+9x^{\frac{3}{5}}=0$

Put $x^{\frac{3}{5}}=t$, we get

$1+8t^2+9t=08t^2+8t+t+1=08t(t+1)+(t+1)=0(8t+1)(t+1)=0t=-1,-\frac{1}{8}{x}^{\frac{3}{5}}=t\Rightarrow x=t^{\frac{5}{3}}$

$x={(-1)}^{\frac{5}{3}},{(-\frac{1}{8})}^{\frac{5}{3}}$

$x=-1,{\left({(-\frac{1}{2})}^3\right)}^{\frac{5}{3}}$

$x=-1,{(-\frac{1}{2})}^{5}x=-1,-\frac{1}{32}$