The expression is given as,
[tex]\sqrt[]{180}[/tex]Use the prime factorization of 180 as,
[tex]\begin{gathered} 180=2\cdot2\cdot3\cdot3\cdot5 \\ 180=2^2\cdot3^2\cdot5 \end{gathered}[/tex]Consider the formulae,
[tex]\begin{gathered} \sqrt[]{x}=(x)^{\frac{1}{2}} \\ (xy)^m=x^my^m \\ (x^m)^n=x^{mn} \end{gathered}[/tex]Then the given expression can be resolved as follows,
[tex]\begin{gathered} \sqrt[]{180} \\ =(180)^{\frac{1}{2}} \\ =(2^2\cdot3^2\cdot5)^{\frac{1}{2}} \\ =(2^2)^{\frac{1}{2}}(3^2)^{\frac{1}{2}}(5)^{\frac{1}{2}} \\ =(2^{2\cdot\frac{1}{2}})^{}(3^{2\cdot\frac{1}{2}})^{}(5)^{\frac{1}{2}} \\ =(2^1)^{}(3^1)^{}\cdot\sqrt[]{5} \\ =6\sqrt[]{5} \end{gathered}[/tex]Thus, the simplest form of the given expression is,
[tex]6\sqrt[]{5}[/tex]