Answer
$25$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents and the definition of rational exponents to simplify the given expression, $
\dfrac{125^{7/3}}{125^{5/3}}
.$
$\bf{\text{Solution Details:}}$
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}
125^{\frac{7}{3}-\frac{5}{3}}
\\\\=
125^{\frac{2}{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}
\left( \sqrt[3]{125} \right)^{2}
\\\\=
\left( \sqrt[3]{(5)^3} \right)^{2}
\\\\=
\left( 5 \right)^{2}
\\\\=
25
.\end{array}
