Answer
$\dfrac{1}{x^{10/3}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents to simplify the given expression, $
\dfrac{\left( x^{2/3}\right)^{2}}{\left( x^2 \right)^{7/3}}
.$
$\bf{\text{Solution Details:}}$
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{x^{\frac{2}{3}\cdot2}}{x^{2\cdot\frac{7}{3}}}
\\\\=
\dfrac{x^{\frac{4}{3}}}{x^{\frac{14}{3}}}
.\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}
x^{\frac{4}{3}-\frac{14}{3}}
\\\\=
x^{-\frac{10}{3}}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{x^{\frac{10}{3}}}
\\\\=
\dfrac{1}{x^{10/3}}
.\end{array}
