Answer
The solutions are $x=32$ and $x=-243$
Work Step by Step
$x^{2/5}+x^{1/5}-6=0$
Let $u$ be equal to $x^{1/5}$
If $u=x^{1/5}$, then $u^{2}=x^{2/5}$
Rewrite the original equation using the new variable $u$:
$u^{2}+u-6=0$
Solve by factoring:
$(u+3)(u-2)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u+3=0$
$u=-3$
$u-2=0$
$u=2$
Substitute $u$ back to $x^{1/5}$ and solve for $x$:
$x^{1/5}=-3$
$(x^{1/5})^{5}=(-3)^{5}$
$x=-243$
$x^{1/5}=2$
$(x^{1/5})^{5}=2^{5}$
$x=32$
The solutions are $x=32$ and $x=-243$
