Sure, let's simplify the cube root of 24 step by step.
1. Prime Factorization:
First, we perform the prime factorization of 24.
[tex]\[
24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3
\][/tex]
2. Identify and Separate Cubic Factors:
Notice that [tex]\(2^3\)[/tex] is a perfect cube. This allows us to separate the cube root:
[tex]\[
\sqrt[3]{24} = \sqrt[3]{2^3 \times 3}
\][/tex]
3. Simplify the Cube Roots:
Using the property of radicals that [tex]\(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\)[/tex], we can rewrite:
[tex]\[
\sqrt[3]{2^3 \times 3} = \sqrt[3]{2^3} \times \sqrt[3]{3}
\][/tex]
4. Simplify:
The cube root of [tex]\(2^3\)[/tex] simplifies to 2:
[tex]\[
\sqrt[3]{2^3} = 2
\][/tex]
So we have:
[tex]\[
2 \times \sqrt[3]{3}
\][/tex]
Thus, the simplified radical form of [tex]\(\sqrt[3]{24}\)[/tex] is:
[tex]\[
2 \sqrt[3]{3}
\][/tex]
