Question

How do you evaluate `\sqrt { \root[ 3] { x } } \cdot \root [ 6] { x ^ { - 1} } \cdot ( x ^ { 2} ) ^ { 2/ 3}`?

Answer 1

`sqrt(root(3)(x)) * root(6)(x^(-1)) * (x^2)^(2/3)= color(red)(x^(4/3)` or `color(red)(xroot(3)(x))`

Explanation:

`color(blue)sqrt(root(3)(x))=(x^(1/3))^(1/2)=color(blue)(x^(1/6))`

`color(green)root(6)(x^(-1))=root(6)(1/x)=1/(root(6)(x))=color(green)(1/x^(1/6))`

`color(brown)(""(x^2)^(2/3))=color(brown)(x^(4/3))`

Therefore
`color(blue)sqrt(root(3)(x)) * color(green)root(6)(x^(-1)) * color(brown)(""(x^2)^(2/3))=color(blue)(cancel(x^(1/6))) * color(green)(1/cancel(x^(1/6))) * color(brown)(x^(4/3))`

`color(white)("XXXXXXXX")=color(brown)(x^(4/3))=x^(3/3) * x^(1/3) = x * root(3)(x)`