Answer
$\text{The given expression simplifies to }\dfrac{1}{x^{10/3}}$
Work Step by Step
$\begin{array}{ l c }
=\dfrac{\left( x^{2/3}\right)^{2}}{\left( x^{7/3}\right)^{2}} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
\left( a^{m}\right)^{n} =\left( a^{n}\right)^{m}
\end{array}\\
& \\
=\left(\dfrac{x^{2/3}}{x^{7/3}}\right)^{2} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
\dfrac{a^{n}}{b^{n}} =\left(\dfrac{a}{b}\right)^{n}
\end{array}\\
& \\
=\left(\dfrac{1}{x^{7/3} \cdot x^{-2/3}}\right)^{2} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
a^{n} =\dfrac{1}{a^{-n}}
\end{array}\\
& \\
=\left(\dfrac{1}{x^{7/3-2/3}}\right)^{2} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
a^{m} \cdot a^{-n} =a^{m-n}
\end{array}\\
& \\
=\left(\dfrac{1}{x^{5/3}}\right)^{2} & \mathrm{Subtract\ exponents}\\
& \\
=\dfrac{1^{2}}{\left( x^{5/3}\right)^{2}} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
\left(\dfrac{a}{b}\right)^{n} =\dfrac{a^{n}}{b^{n}}
\end{array}\\
& \\
=\dfrac{1}{x^{( 5/3) \cdot ( 2)}} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
\left( a^{m}\right)^{n} =a^{m\cdot m}
\end{array}\\
& \\
=\dfrac{1}{x^{10/3}} & \mathrm{Simplify}
\end{array}$
