Example  2
Simplify the expression \(\frac{{6\sqrt[3]{{48}}}}{{2\sqrt[3]{3}}}\).


Step 1: Divide the coefficients and divide the radicands.

\[\begin{align}
 \frac{{6\sqrt[3]{{48}}}}{{2\sqrt[3]{3}}} &= \left( {\frac{6}{2}} \right)\sqrt[3]{{\frac{{48}}{3}}} \\ 
  &= 3\sqrt[3]{{16}} \\ 
 \end{align}\]


Step 2: Check to see if the radical can be further simplified.

\[\begin{align}
 3\sqrt[3]{{16}} &= 3\sqrt[3]{{2^4 }} \\ 
  &= 3\sqrt[3]{{2^3 \cdot 2^1 }} \\ 
  &= \left( {3\cdot 2} \right)\sqrt[3]{2} \\ 
  &= 6\sqrt[3]{2} \\ 
 \end{align}\]

Therefore, \(\frac{{6\sqrt[3]{{48}}}}{{2\sqrt[3]{3}}} = 6\sqrt[3]{2}\).