Answer
$\dfrac{-25\sqrt[3]{9}}{18}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Simplify each term in the given expression, $ \dfrac{-4}{\sqrt[3]{3}}+\dfrac{1}{\sqrt[3]{24}}-\dfrac{2}{\sqrt[3]{81}} .$ Then make the terms similar (same denominator) to combine the numerators.
$\bf{\text{Solution Details:}}$
Extracting the perfect cube factors of the radicals results to \begin{array}{l}\require{cancel} \dfrac{-4}{\sqrt[3]{3}}+\dfrac{1}{\sqrt[3]{8\cdot3}}-\dfrac{2}{\sqrt[3]{27\cdot3}} \\\\= \dfrac{-4}{\sqrt[3]{3}}+\dfrac{1}{\sqrt[3]{(2)^3\cdot3}}-\dfrac{2}{\sqrt[3]{(3)^3\cdot3}} \\\\= \dfrac{-4}{\sqrt[3]{3}}+\dfrac{1}{2\sqrt[3]{3}}-\dfrac{2}{3\sqrt[3]{3}} .\end{array} Rationalizing the denominators results to \begin{array}{l}\require{cancel} \dfrac{-4}{\sqrt[3]{3}}\cdot\dfrac{\sqrt[3]{3^2}}{\sqrt[3]{3^2}}+\dfrac{1}{2\sqrt[3]{3}}\cdot\dfrac{\sqrt[3]{3^2}}{\sqrt[3]{3^2}}-\dfrac{2}{3\sqrt[3]{3}}\cdot\dfrac{\sqrt[3]{3^2}}{\sqrt[3]{3^2}} \\\\= \dfrac{-4\sqrt[3]{3^2}}{\sqrt[3]{3^3}}+\dfrac{\sqrt[3]{3^2}}{2\sqrt[3]{3^3}}-\dfrac{2\sqrt[3]{3^2}}{3\sqrt[3]{3^3}} \\\\= \dfrac{-4\sqrt[3]{3^2}}{3}+\dfrac{\sqrt[3]{3^2}}{2(3)}-\dfrac{2\sqrt[3]{3^2}}{3(3)} \\\\= \dfrac{-4\sqrt[3]{3^2}}{3}+\dfrac{\sqrt[3]{3^2}}{6}-\dfrac{2\sqrt[3]{3^2}}{9} \\\\= \dfrac{-4\sqrt[3]{9}}{3}+\dfrac{\sqrt[3]{9}}{6}-\dfrac{2\sqrt[3]{9}}{9} .\end{array}
To simplify the expression above, find the $LCD$ of the denominators. The $LCD$ is $18$ since it is the lowest number that can be exactly divided by the denominators $
3,6, \text{ and } 9
.$ Multiplying each term by an expression equal to $1$ so that its denominator becomes the $LCD$ results to
\begin{array}{l}\require{cancel} \dfrac{-4\sqrt[3]{9}}{3}\cdot\dfrac{6}{6}+\dfrac{\sqrt[3]{9}}{6}\cdot\dfrac{3}{3}-\dfrac{2\sqrt[3]{9}}{9}\cdot\dfrac{2}{2} \\\\= \dfrac{-24\sqrt[3]{9}}{18}+\dfrac{3\sqrt[3]{9}}{18}-\dfrac{4\sqrt[3]{9}}{18} .\end{array} To combine similar terms, add/subtract the numerators and copy the similar denominator. Hence, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{-24\sqrt[3]{9}+3\sqrt[3]{9}-4\sqrt[3]{9}}{18} \\\\=
\dfrac{-25\sqrt[3]{9}}{18}
.\end{array}
