## Intermediate Algebra (12th Edition)

$4$
$\bf{\text{Solution Outline:}}$ Use the laws of exponents and the definition of rational exponents to simplify the given expression, $\dfrac{64^{5/3}}{64^{4/3}} .$ $\bf{\text{Solution Details:}}$ 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} 64^{\frac{5}{3}-\frac{4}{3}} \\\\= 64^{\frac{1}{3}} .\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 expression above is equivalent to \begin{array}{l}\require{cancel} \sqrt[3]{64^1} \\\\= \sqrt[3]{64} \\\\= \sqrt[3]{(4)^3} \\\\= 4 .\end{array}