## Intermediate Algebra (6th Edition)

$\dfrac{15|x|\sqrt{2x}}{2}$
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the given expression, $\dfrac{3\sqrt{100x^2}}{2\sqrt{2x^{-1}}} ,$ is equivalent to \begin{array}{l}\require{cancel} \dfrac{3}{2}\sqrt{\dfrac{100x^2}{2x^{-1}}} \\= \dfrac{3}{2}\sqrt{\dfrac{50x^2}{x^{-1}}} .\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} \dfrac{3}{2}\sqrt{50x^{2-(-1)}} \\= \dfrac{3}{2}\sqrt{50x^{2+1}} \\= \dfrac{3}{2}\sqrt{50x^{3}} .\end{array} Extracting the factor of the radicand that is a perfect power of the index, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{3}{2}\sqrt{50x^{3}} \\= \dfrac{3}{2}\sqrt{25x^{2}\cdot2x} \\= \dfrac{3}{2}\sqrt{(5x)^{2}\cdot2x} .\end{array} Using $\sqrt[n]{x^n}=|x|$ if $n$ is even and $\sqrt[n]{x^n}=x$ if $n,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{3}{2}|5x|\sqrt{2x} \\\\= \dfrac{3}{2}\cdot5|x|\sqrt{2x} \\\\= \dfrac{15|x|\sqrt{2x}}{2} .\end{array}