Answer
$\frac{1}{x^{\frac{11}{12}}}$
Work Step by Step
We are given the expression $\frac{x^{\frac{1}{4}}x^{-\frac{1}{2}}}{x^{\frac{2}{3}}}$.
We can use the product rule to simplify, which holds that $a^{m}\times a^{n}=a^{m+n}$ (where a is a real number, and m and n are positive integers).
$\frac{x^{\frac{1}{4}+(-\frac{1}{2})}}{x^{\frac{2}{3}}}=\frac{x^{\frac{1}{4}+(-\frac{2}{4})}}{x^{\frac{2}{3}}}=\frac{x^{-\frac{1}{4}}}{x^{\frac{2}{3}}}$
Next, we can use the quotient rule, which holds that $\frac{a^{m}}{a^{n}}=a^{m-n}$ (where a is a nonzero real number, and m and n are integers).
$x^{-\frac{1}{4}-\frac{2}{3}}=x^{-\frac{3}{12}-\frac{8}{12}}=x^{-\frac{11}{12}}=\frac{1}{x^{\frac{11}{12}}}$
