Question

Which expression is a perfect cube?

Answer 1

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

$64x^{15} y^{18}$

Answer 2

The expression that is a perfect cube is D. Because to determine the side length of a cube we would need to take the cube root of the number and the variables and unlike the other answers we can't take the cube root out perfectly. For D though the side length is calculated by taking the cube root of 64, which is 4, cube root of x^27 which is x^3, and cube root of y^125 which is y^5. Putting everything together the side length should be 4X^3Y^5. Multiplying this expression by itself 3 times will give you what you started out with.