Answer: the fourth expression: [tex]64x^6y^{12}+125x^9y^3[/tex]
Justification:
1) Find wich terms have exact cubic root
2) Given that 32 does not have exact cubic root, the second and third expressions can be discarded.
3) The first expression has a term with negative signs, so its cubic root will keep the negative signs, which will result a difference and not a sum of cubic terms.
4) That leads to the fourth expression, which I will analyze in detail:
First term: [tex] \sqrt[3]{64x^6y^{12}} =4x^2y^4[/tex]
=> [tex](4x^2y^4)^{3}=64x^6y^{12}[/tex]
Second term: [tex] \sqrt[3]{125x^9y^3}=5x^3y [/tex]
=> [tex](5x^3y)^{3}=125x^9y^3[/tex]
So, the fourth expression it the sum of two cubes: [tex] (4x^2y^4)^{3} + (5x^3y)^{3} [/tex]
