Question

Express √180 as a mixed radical in simplest form

Answer 1

The expression is given as,

`\sqrt[]{180}`

Use the prime factorization of 180 as,

`\begin{gathered} 180=2\cdot2\cdot3\cdot3\cdot5 \\ 180=2^2\cdot3^2\cdot5 \end{gathered}`

Consider the formulae,

`\begin{gathered} \sqrt[]{x}=(x)^{(1)/(2)} \\ (xy)^m=x^my^m \\ (x^m)^n=x^(mn) \end{gathered}`

Then the given expression can be resolved as follows,

`\begin{gathered} \sqrt[]{180} \\ =(180)^{(1)/(2)} \\ =(2^2\cdot3^2\cdot5)^{(1)/(2)} \\ =(2^2)^{(1)/(2)}(3^2)^{(1)/(2)}(5)^{(1)/(2)} \\ =(2^{2\cdot(1)/(2)})^{}(3^{2\cdot(1)/(2)})^{}(5)^{(1)/(2)} \\ =(2^1)^{}(3^1)^{}\cdot\sqrt[]{5} \\ =6\sqrt[]{5} \end{gathered}`

Thus, the simplest form of the given expression is,

`6\sqrt[]{5}`