we know that
A perfect cube is a whole number which is the cube of another whole number
so
A number is a perfect cube, if the cubic root of the number is equal to a whole number
case 1)
[tex] 9a^{3} - 27[/tex]
[tex] 9a^{3} - 27 = 9a^{3}-3 ^{3} [/tex]
in this problem the cubic root of [tex] 9 [/tex] is not a whole number
therefore
The case 1) is not a differences of perfect cubes
case 2)
[tex] 125a^{12} - 72 [/tex]
[tex] 125a^{12} - 72=5^{3}(a^{4})^{3} -\sqrt[3]{72} [/tex]
in this problem the cubic root of [tex] 72 [/tex] is not a whole number
therefore
The case 2) is not a differences of perfect cubes
case 3)
[tex] 216a^{6}- 27y^{3} [/tex]
[tex] 216a^{6}- 27y^{3} = 6^{3}(a^{2})^{3}- 3^{3}y^{3} = (6a^{2})^{3} - (3y)^{3}[/tex]
therefore
The case 3) is a differences of perfect cubes
case 4)
[tex] 8a^{15} - 27 [/tex]
[tex] 8a^{15} - 27 = 2^{3}( a^{5})^{3} - 3^{3} = (2a^{5} )^{3} - 3^{3} [/tex]
therefore
The case 4) is a differences of perfect cubes
case 5)
[tex] 225a^{3} - 216 [/tex]
[tex] 225a^{3} - 216 =5^{3}a^{3} -6^{3}= (5a)^{3} - 6^{3} [/tex]
therefore
The case 5) is a differences of perfect cubes
therefore
the answer is
[tex] 216a^{6}- 27y^{3} [/tex]
[tex] 8a^{15} - 27 [/tex]
[tex] 225a^{3} - 216 [/tex]
