Answer
$\color{blue}{z^{1/3}}$
Work Step by Step
RECALL:
(1) $a^m \cdot a^n = a^{m+n}$
(2) $\dfrac{a^m}{a^n} = a^{m-n}$
(3) $a^{1/n} = \sqrt[n]{a}$
(4) When $n$ is odd, $\sqrt[n]{a^n}=a$
(5) $a^{m/n} = \left(\sqrt[n]{a}\right)^m$
(6) $(ab)^m=a^mb^m$
(7) $(a^m)^n=a^{mn}$
(8) $a^{-m}=\dfrac{1}{a^m}$
Use rule (1) above to obtain:
$=\dfrac{z^{1/3-2/3+1/6}}{(z^{-1/6})^3}
\\=\dfrac{z^{2/6-4/6+1/6}}{(z^{-1/6})^{3}}
\\=\dfrac{z^{-1/6}}{(z^{-1/6})^{3}}$
Use rule (7) above to obtain:
$\\=\dfrac{z^{-1/6}}{z^{(-1/6) \cdot 3}}
\\=\dfrac{z^{-1/6}}{z^{-3/6}}$
Use rule (2) above to obtain:
$=z^{-1/6-(-3/6)}
\\=z^{-1/6+3/6}
\\=z^{2/6}
\\=\color{blue}{z^{1/3}}$
