Answer
$\sqrt[6]{7}$
Work Step by Step
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{7^{-1/3}}{7^{-1/2}}
\\\\=
7^{-\frac{1}{3}-\left( -\frac{1}{2} \right)}
\\\\=
7^{-\frac{1}{3}+\frac{1}{2}}
\\\\=
7^{-\frac{2}{6}+\frac{3}{6}}
\\\\=
7^{\frac{1}{6}}
.\end{array}
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the given expression, $
8^{2/3}
,$ is equivalent to
\begin{array}{l}\require{cancel}
7^{\frac{1}{6}}
\\\\=
\sqrt[6]{7^1}
\\\\=
\sqrt[6]{7}
.\end{array}
