#### Answer

$(-4,5]$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
The domain of the given function, $
g(x)=\dfrac{\sqrt[4]{5-x}}{\sqrt[6]{x+4}}
,$ are the combined restrictions of the numerator and the denominator.
$\bf{\text{Solution Details:}}$
In the numerator, since the radicand of a radical with an even index (index equals $
4
$), should be nonnegative, then
\begin{array}{l}\require{cancel}
5-x\ge0
\\\\
-x\ge-5
\\\\
x\le\dfrac{-5}{-1}
\\\\
x\le5
.\end{array}
In the denominator, since the radicand of a radical with an even index (index equals $
6
$), should be nonnegative, and that the denominator cannot be zero, then
\begin{array}{l}\require{cancel}
x+4\gt0
\\\\
x\gt-4
.\end{array}
Combining the two restrictions above, then the domain is the interval $
(-4,5]
.$