Answer
$a+2a\sqrt[3]{3}$
Work Step by Step
Using $a(b+c)=ab+ac$ or the Distributive Property and the properties of radicals, the given expression, $
\sqrt[3]{a}(\sqrt[3]{a^2}+\sqrt[3]{24a^2})
,$ is equivalent to
\begin{array}{l}\require{cancel}
(\sqrt[3]{a})(\sqrt[3]{a^2})+(\sqrt[3]{a})(\sqrt[3]{24a^2})
\\\\=
\sqrt[3]{a(a^2)}+\sqrt[3]{a(24a^2)}
\\\\=
\sqrt[3]{a^3}+\sqrt[3]{24a^3}
\\\\=
\sqrt[3]{a^3}+\sqrt[3]{8a^3\cdot3}
\\\\=
\sqrt[3]{(a)^3}+\sqrt[3]{(2a)^3\cdot3}
\\\\=
a+2a\sqrt[3]{3}
.\end{array}
