To simplified the expression we need to write the radicand as product of powers that are multiples of three. Let's do that:
[tex]\sqrt[3]{81y^7}=\sqrt[3]{3^3\cdot3\cdot y^6\cdot y}[/tex]
Now we grouped the terms in powers we can take the root of:
[tex]\begin{gathered} \sqrt[3]{81y^7}=\sqrt[3]{3^3\cdot3\cdot y^6\cdot y} \\ =\sqrt[3]{(3^3y^6)(3y)} \\ =\sqrt[3]{(3y^2)^3(3y)} \end{gathered}[/tex]
Finally we perform the root:
[tex]\begin{gathered} \sqrt[3]{81y^7}=\sqrt[3]{3^3\cdot3\cdot y^6\cdot y} \\ =\sqrt[3]{(3^3y^6)(3y)} \\ =\sqrt[3]{(3y^2)^3(3y)} \\ =3y^2\sqrt[3]{3y} \end{gathered}[/tex]
Therefore:
[tex]\sqrt[3]{81y^7}=3y^2\sqrt[3]{3y}[/tex]
