Simplify the radical expression: the cube root of 4 times the cube root of 6.

Step-by-step solution

1. 1

   Combine under one radical. Radicals of the same index multiply directly: $\sqrt[3]{4}\cdot\sqrt[3]{6} = \sqrt[3]{24}$.

2. 2

   Hunt for a perfect-cube factor. Factor $24 = 2^3 \cdot 3$. The $2^3 = 8$ is a perfect cube; the leftover $3$ is not.

3. 3

   Split the radical. $\sqrt[3]{24} = \sqrt[3]{8}\cdot\sqrt[3]{3}$.

4. 4

   Extract the cube. $\sqrt[3]{8} = 2$, leaving $2\sqrt[3]{3}$. Since $3$ has no cube factors, the expression is fully simplified.

5. 5

   Check numerically. $\sqrt[3]{4} \approx 1.587401$ and $\sqrt[3]{6} \approx 1.817121$, whose product is $\approx 2.884499$; and $2\sqrt[3]{3} \approx 2 \cdot 1.442250 = 2.884499$.