[tex] \begin{align} \sqrt[3] {108} &= \sqrt[3] {4×27} \\ &= \sqrt[3] {4×3^3} \\ &= 3\sqrt [3]{4}\end{align} [/tex]
[tex] \begin{align} \sqrt {\dfrac{2}{3}} &= \sqrt {\dfrac{2}{3}} \cdot \dfrac{\sqrt 3}{\sqrt 3} \\ &= \dfrac{\sqrt 2\cdot \sqrt 3}{\sqrt 3\cdot \sqrt 3} \\ &= \dfrac{\sqrt {6}}{3}\end{align} [/tex]
[tex] \begin{align} \sqrt[3]{\dfrac{1}{2}} &= \dfrac{1}{\sqrt[3]{2}} \cdot \dfrac{\sqrt[3]{4}}{\sqrt[3]{4}} \\ &= \dfrac{\sqrt[3]{4}}{\sqrt[3]{2}\cdot\sqrt[3]{4}} \\ &= \dfrac{\sqrt[3]{4}}{2}\end{align} [/tex]
[tex] \begin{align} \sqrt{\dfrac{5}{9}} &= \dfrac{\sqrt5}{\sqrt9} \\ &= \dfrac{\sqrt5}{3}\end{align} [/tex]
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[tex] \begin{align} \sqrt[3] {108} &= \sqrt[3] {4×27} \\ &= \sqrt[3] {4×3^3} \\ &= 3\sqrt [3]{4}\end{align} [/tex]
[tex] \begin{align} \sqrt {\dfrac{2}{3}} &= \sqrt {\dfrac{2}{3}} \cdot \dfrac{\sqrt 3}{\sqrt 3} \\ &= \dfrac{\sqrt 2\cdot \sqrt 3}{\sqrt 3\cdot \sqrt 3} \\ &= \dfrac{\sqrt {6}}{3}\end{align} [/tex]
[tex] \begin{align} \sqrt[3]{\dfrac{1}{2}} &= \dfrac{1}{\sqrt[3]{2}} \cdot \dfrac{\sqrt[3]{4}}{\sqrt[3]{4}} \\ &= \dfrac{\sqrt[3]{4}}{\sqrt[3]{2}\cdot\sqrt[3]{4}} \\ &= \dfrac{\sqrt[3]{4}}{2}\end{align} [/tex]
[tex] \begin{align} \sqrt{\dfrac{5}{9}} &= \dfrac{\sqrt5}{\sqrt9} \\ &= \dfrac{\sqrt5}{3}\end{align} [/tex]