we know that
A perfect cube is a whole number which is the cube of another whole number or a number is a perfect cube if the cube root of that number is a whole number.
so
case a) [tex] 12x^{12} y^{18}[/tex]
[tex] 12x^{12} y^{18} = 12(x^{4} y^{6})^{3}[/tex]
the cube root of [tex] 12 [/tex] is not a whole number
therefore
The expression is not a perfect cube
case b) [tex] 12x^{27} y^{125}[/tex]
[tex] 12x^{27} y^{125} = 12(x^{9})^{3}y^{125} [/tex]
the cube root of [tex] 12 [/tex] is not a whole number
the cube root of [tex] y^{125} [/tex] is not a whole number
therefore
The expression is not a perfect cube
case c) [tex] 64x^{15} y^{18}[/tex]
[tex] 64x^{15} y^{18} = (4 x^{5} y^{6})^{3}[/tex]
therefore
The expression is a perfect cube
case d) [tex] 64x^{27} y^{125}[/tex]
[tex] 64x^{27} y^{125} = (4 x^{9})^{3}y^{125}[/tex]
the cube root of [tex] y^{125} [/tex] is not a whole number
therefore
The expression is not a perfect cube
the answer is
[tex] 64x^{15} y^{18}[/tex]
