Answer
The solutions are $x=5$ and $x=-4$
Work Step by Step
$(x^{2}-x-4)^{3/4}-2=6$
Take the $2$ to the right side of the equation:
$(x^{2}-x-4)^{3/4}=6+2$
$(x^{2}-x-4)^{3/4}=8$
Raise both sides to $\dfrac{4}{3}$:
$[(x^{2}-x-4)^{3/4}]^{4/3}=8^{4/3}$
$x^{2}-x-4=\sqrt[3]{8^{4}}$
Simplify the right side:
$x^{2}-x-4=\sqrt[3]{8^{3}\cdot8}$
$x^{2}-x-4=(8)(2)$
$x^{2}-x-4=16$
Take $16$ to the left side and simplify:
$x^{2}-x-4-16=0$
$x^{2}-x-20=0$
Solve by factoring:
$(x-5)(x+4)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$x-5=0$
$x=5$
$x+4=0$
$x=-4$
The solutions are $x=5$ and $x=-4$