Example 2

Method 1: Prime factorization

Step 1: Rewrite the radicand as a product of prime factors.

\(\sqrt[3]{{48}} = \sqrt[3]{{2 \cdot 2\cdot 2\cdot 2\cdot 3}}\)

Step 2: Group as many identical factors as match the value of the index. Because the index is \(3\), group three factors of two, \(2 \cdot 2 \cdot 2\), and rewrite them as \(2^3\), a perfect cube.

\(\sqrt[3]{{{\color{red}2 \cdot 2\cdot 2}\cdot 2\cdot 3}} = \sqrt[3]{{2^3 \cdot 2\cdot 3}}\)

Step 3: Rewrite the radical as a product of the perfect cube radical and the non-perfect cube radicals.

\(\sqrt[3]{{2^3 \cdot 2 \cdot 3}} = \sqrt[3]{{2^3}} \cdot \sqrt[3]{{2\cdot 3}}\)

Step 4: Simplify.

\(\sqrt[3]{{2^3}}\cdot \sqrt[3]{{2\cdot 3}} = 2 \cdot \sqrt[3]{6}\)

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

Method 2: Perfect cubes (because the index is 3)

Step 1: Rewrite the radicand as a product of factors, looking specifically for factors that are perfect cubes.

\(\begin{align}
 \sqrt[3]{{48}} &= \sqrt[3]{{8\cdot 6}} \\ 
  &= \sqrt[3]{{2^3 \cdot 6}} \\ 
 \end{align}\)

Step 2: Rewrite the radical as a product of the perfect cube radical and the non-perfect cube radicals.

\(\sqrt[3]{{2^3 \cdot 6}} = \sqrt[3]{{2^3}} \cdot \sqrt[3]{6}\)

Step 3: Simplify.

\(\sqrt[3]{{2^3}}\cdot \sqrt[3]{6} = 2\sqrt[3]{6}\)

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