## Intermediate Algebra (12th Edition)

$\dfrac{1}{x^{10/3}}$
$\bf{\text{Solution Outline:}}$ 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}