## Intermediate Algebra (12th Edition)

$\dfrac{1}{p^{7/4}}$
$\bf{\text{Solution Outline:}}$ Use the laws of exponents to simplify the given expression, $\dfrac{\left( p^{3}\right)^{1/4}}{\left( p^{5/4} \right)^{2}} .$ $\bf{\text{Solution Details:}}$ 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{p^{3\cdot\frac{1}{4}}}{p^{\frac{5}{4}\cdot2}} \\\\= \dfrac{p^{\frac{3}{4}}}{p^{\frac{10}{4}}} .\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} p^{\frac{3}{4}-\frac{10}{4}} \\\\= p^{-\frac{7}{4}} .\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}{p^{\frac{7}{4}}} \\\\= \dfrac{1}{p^{7/4}} .\end{array}