Answer
The solutions are $x=-8$ and $x=27$
Work Step by Step
$x^{2/3}-x^{1/3}-6=0$
Let $u$ be equal to $x^{1/3}$
If $u=x^{1/3}$, then $u^{2}=x^{2/3}$
Rewrite the given 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 $x^{1/3}$ and solve for $x$:
$u=3$
$x^{1/3}=3$
$(x^{1/3})^{3}=3^{3}$
$x=27$
$u=-2$
$x^{1/3}=-2$
$(x^{1/3})^{3}=(-2)^{3}$
$x=-8$
The solutions are $x=-8$ and $x=27$
