Answer
$( y-6 )( y^2+6y+36)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
y^3-216
,$ use the factoring of the difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
The expressions $
y^3
$ and $
216
$ are both perfect cubes (the cube root is exact). Hence, $
y^3-216
,$ is a difference of $2$ cubes. 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}
( y)^3-(6 )^3
\\\\=
( y-6 )[( y)^2+ y(6 )+(6 )^2]
\\\\=
( y-6 )( y^2+6y+36)
.\end{array}
