Questions: Simplify the radical ∛54 : □

Transcript text: Simplify the radical $\sqrt[3]{54}$ : $\square$

Solution

Solution Steps

To simplify the radical $\sqrt[3]{54}$, we need to find the prime factorization of 54 and then simplify the cube root by grouping the factors into sets of three.

Step 1: Prime Factorization

To simplify $\sqrt[3]{54}$, we first find the prime factorization of 54. We can express 54 as: \[ 54 = 2 \times 3^3 \]

Step 2: Simplifying the Cube Root

Next, we apply the property of cube roots to simplify: \[ \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = \sqrt[3]{2} \times \sqrt[3]{3^3} \] Since $\sqrt[3]{3^3} = 3$, we can rewrite the expression as: \[ \sqrt[3]{54} = 3 \times \sqrt[3]{2} \]

Step 3: Final Expression

Thus, the simplified form of $\sqrt[3]{54}$ is: \[ \sqrt[3]{54} = 3 \cdot 2^{1/3} \]