Answer
$10$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(2+\sqrt[3]{2})(4-2\sqrt[3]{2}+\sqrt[3]{4})
,$ use the factoring of 2 cubes.
$\bf{\text{Solution Details:}}$
Using the factoring of the sum or difference of $2$ cubes which is given by $a^3+b^3=(a+b)(a^2-ab+b^2)$ or by $a^3-b^3=(a-b)(a^2+ab+b^2)$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(2+\sqrt[3]{2})[(2)^2-2(\sqrt[3]{2})+(\sqrt[3]{2})^2]
\\\\=
(2)^3+(\sqrt[3]{2})^2
\\\\=
8+2
\\\\=
10
.\end{array}
