## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$(-4,5]$
$\bf{\text{Solution Outline:}}$ 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 $(-4,5] .$