Questions: Simplify the radical expression. ∛4 ⋅ ∛6 ∛4 ⋅ ∛6 =

Transcript text: N9: Radical Expressions Simplify the radical expression. \[ \begin{array}{r} \sqrt[3]{4} \cdot \sqrt[3]{6} \\ \sqrt[3]{4} \cdot \sqrt[3]{6}= \end{array} \] $\square$

Solution

Solution Steps

To simplify the given radical expression, we can use the property of radicals that states $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$. In this case, we have cube roots, so we can combine the terms under a single cube root.

Step 1: Combine the Cube Roots

We start with the expression: \[ \sqrt[3]{4} \cdot \sqrt[3]{6} \] Using the property of radicals, we can combine these into a single cube root: \[ \sqrt[3]{4 \cdot 6} = \sqrt[3]{24} \]

Step 2: Simplify the Expression

Next, we simplify $\sqrt[3]{24}$. We can factor $24$ into its prime factors: \[ 24 = 2^3 \cdot 3 \] Thus, we can express the cube root as: \[ \sqrt[3]{24} = \sqrt[3]{2^3 \cdot 3} = \sqrt[3]{2^3} \cdot \sqrt[3]{3} = 2 \cdot \sqrt[3]{3} \]

Step 3: Final Expression

The simplified form of the original expression is: \[ 2 \cdot \sqrt[3]{3} \]