X raised to a fract. exponent, raised to another fract. exponent

(x^(3/2))^(1/6)

___________

(x^(2/3))^(1/6)

simplify and use only positive exponents

$(x^\frac{3}{2})^\frac{1}{6}/(x^\frac{2}{3})^\frac{1}{6}$

$=x^\frac{1}{4}/x^\frac{1}{9}\\ =x^{\frac{1}{4}-\frac{1}{9}}\\ \color{blue}=x^{\frac{5}{36}}$ 

  !

(x^(3/2))^(1/6)

___________

(x^(2/3))^(1/6)

simplify and use only positive exponents

$\begin{array}{|rcll|} \hline && \dfrac{ (x^{\frac{3}{2}} )^{ \frac{1}{6} } } { (x^{\frac{2}{3}})^{ \frac{1}{6} } } \\ &=& \left(\dfrac{ x^{\frac{3}{2}} } { x^{\frac{2}{3}} } \right)^{ \frac{1}{6} } \\ &=& \left(x^{\frac{3}{2}} \cdot x^{-\frac{2}{3}} \right)^{ \frac{1}{6} } \\ &=& \left(x^{\frac{3}{2}-\frac{2}{3}} \right)^{ \frac{1}{6} } \\ &=& \left(x^{\frac{9-4}{6}} \right)^{ \frac{1}{6} } \\ &=& \left(x^{\frac{5}{6}} \right)^{ \frac{1}{6} } \\ &=& x^{\frac{5}{6}\cdot\frac{1}{6} } \\ &=& x^{\frac{5}{36}} \\ \hline \end{array}$