Answer: The answer is (D) [tex]64x^6.[/tex]
Step-by-step explanation: We are to select the correct monomial from the given options which is a perfect cube.
The first option is
[tex]y=16x^6=2\times (8x^6)=2\times(2x^2)^3\\\\\Rightarrow \sqrt[3]{y}=2x^2\sqrt[3]{2}.[/tex]
So, this is not a perfect cube.
The second option is
[tex]y=27x^8=x^2\times (27x^6)=x^2\times(3x^2)^3\\\\\Rightarrow \sqrt[3]{y}=3x^2\sqrt[3]{x^2}.[/tex]
So, this is not a perfect cube.
The third option is
[tex]y=32x^{12}=4\times (8x^{12})=4\times(2x^4)^3\\\\\Rightarrow \sqrt[3]{y}=2x^4\sqrt[3]{4}.[/tex]
So, this is not a perfect cube.
The fourth option is
[tex]y=64x^6=(4x^2)^3\\\\\Rightarrow \sqrt[3]{y}=4x^2.[/tex]
So, this is a perfect cube.
Thus, (D) [tex]64x^6[/tex] is the correct option.
