Answer
$53\sqrt[3]{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
15\sqrt[3]{81}+4\sqrt[3]{24}
,$ simplify first each term by expressing the radicand as a factor that is a perfect power of the index. Then, extract the root. Finally, combine the like radicals.
$\bf{\text{Solution Details:}}$
Expressing the radicand as an expression that contains a factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
15\sqrt[3]{27\cdot3}+4\sqrt[3]{8\cdot3}
\\\\=
15\sqrt[3]{(3)^3\cdot3}+4\sqrt[3]{(2)^3\cdot3}
.\end{array}
Extracting the roots of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
15(3)\sqrt[3]{3}+4(2)\sqrt[3]{3}
\\\\=
45\sqrt[3]{3}+8\sqrt[3]{3}
.\end{array}
By combining like radicals, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(45+8)\sqrt[3]{3}
\\\\=
53\sqrt[3]{3}
.\end{array}