The given expression is,
[tex]\sqrt[3]{125x^2y^7}[/tex]
Consider the notation,
[tex]\sqrt[m]{x}=x^{\frac{1}{m}}[/tex]
Then the expression becomes,
[tex](125x^2y^7)^{\frac{1}{3}}[/tex]
Consider the following properties of exponents,
[tex]\begin{gathered} (mn)^p=m^pn^p \\ (x^m)^n=x^{mn} \end{gathered}[/tex]
Apply the properties to simplify the expression,
[tex]\begin{gathered} =(125)^{\frac{1}{3}}(x^2)^{\frac{1}{3}}(y^7)^{\frac{1}{3}} \\ =(5^3)^{\frac{1}{3}}(x^2)^{\frac{1}{3}}(y^6\cdot y)^{\frac{1}{3}} \\ =(5^{3\cdot\frac{1}{3}})^{}(x^{2\cdot\frac{1}{3}})^{}(y^{6\cdot\frac{1}{3}}\cdot y^{\frac{1}{3}}) \\ =(5^{})^{}(x^{\frac{2}{3}})^{}(y^2\cdot y^{\frac{1}{3}}) \\ =5y^2(x^2y)^{\frac{1}{3}} \\ =5y^2\cdot\sqrt[3]{x^2y} \end{gathered}[/tex]
Thus, the simplest form of the given expression is obtained as,
[tex]5y^2\cdot\sqrt[3]{x^2y}[/tex]
