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) \(12x^{12} y^{18}\)
\(12x^{12} y^{18} = 12(x^{4} y^{6})^{3}\)
the cube root of \(12\) is not a whole number
therefore
The expression is not a perfect cube
case b) \(12x^{27} y^{125}\)
\(12x^{27} y^{125} = 12(x^{9})^{3}y^{125}\)
the cube root of \(12\) is not a whole number
the cube root of \(y^{125}\) is not a whole number
therefore
The expression is not a perfect cube
case c) \(64x^{15} y^{18}\)
\(64x^{15} y^{18} = (4 x^{5} y^{6})^{3}\)
therefore
The expression is a perfect cube
case d) \(64x^{27} y^{125}\)
\(64x^{27} y^{125} = (4 x^{9})^{3}y^{125}\)
the cube root of \(y^{125}\) is not a whole number
therefore
The expression is not a perfect cube
the answer is
\(64x^{15} y^{18}\)
