Answer
$z^{1/3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents to simplify the given expression, $
\dfrac{ z^{1/3}z^{-2/3}z^{1/6}}{\left( z^{-1/6} \right)^{3}}
.$
$\bf{\text{Solution Details:}}$
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{ z^{1/3}z^{-2/3}z^{1/6}}{z^{-\frac{1}{6}\cdot3}}
\\\\=
\dfrac{ z^{1/3}z^{-2/3}z^{1/6}}{z^{-\frac{3}{6}}}
.\end{array}
Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{ z^{\frac{1}{3}+\left( -\frac{2}{3} \right)+\frac{1}{6}}}{z^{-\frac{3}{6}}}
\\\\=
\dfrac{ z^{\frac{1}{3}-\frac{2}{3}+\frac{1}{6}}}{z^{-\frac{3}{6}}}
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
z^{\frac{1}{3}-\frac{2}{3}+\frac{1}{6}-\left( -\frac{3}{6} \right)}
\\\\
z^{\frac{1}{3}-\frac{2}{3}+\frac{1}{6}+\frac{3}{6}}
\\\\
z^{\frac{2}{6}-\frac{4}{6}+\frac{1}{6}+\frac{3}{6}}
\\\\
z^{\frac{2}{6}}
\\\\
z^{\frac{1}{3}}
\\\\
z^{1/3}
.\end{array}
