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Simplify \(\frac{{\sqrt[3]{4}}}{{2\sqrt[3]{6}}}\).
Check if you can simplify before rationalizing the denominator.
\(\frac{{\sqrt[3]{4}}}{{2\sqrt[3]{6}}} = \frac{{\sqrt[3]{2}}}{{2\sqrt[3]{3}}}\)
Rationalize the denominator.
\(\begin{align}
\frac{{\sqrt[3]{2}}}{{2\sqrt[3]{3}}} &= \frac{{\sqrt[3]{2}}}{{2\sqrt[3]{3}}}\cdot \frac{{\left( {\sqrt[3]{3}} \right)^2 }}{{\left( {\sqrt[3]{3}} \right)^2 }} \\
&= \frac{{\sqrt[3]{{2\cdot 3^2 }}}}{{2\sqrt[3]{{3^3 }}}} \\
&= \frac{{\sqrt[3]{{18}}}}{{2\cdot 3}} \\
&= \frac{{\sqrt[3]{{18}}}}{6} \\
\end{align}\)
\(\frac{{\sqrt[3]{4}}}{{2\sqrt[3]{6}}} = \frac{{\sqrt[3]{2}}}{{2\sqrt[3]{3}}}\)
Rationalize the denominator.
\(\begin{align}
\frac{{\sqrt[3]{2}}}{{2\sqrt[3]{3}}} &= \frac{{\sqrt[3]{2}}}{{2\sqrt[3]{3}}}\cdot \frac{{\left( {\sqrt[3]{3}} \right)^2 }}{{\left( {\sqrt[3]{3}} \right)^2 }} \\
&= \frac{{\sqrt[3]{{2\cdot 3^2 }}}}{{2\sqrt[3]{{3^3 }}}} \\
&= \frac{{\sqrt[3]{{18}}}}{{2\cdot 3}} \\
&= \frac{{\sqrt[3]{{18}}}}{6} \\
\end{align}\)