To determine which expression is equivalent to [tex]\( 24^{\frac{1}{3}} \)[/tex], we can break down the problem step-by-step:
1. Prime Factorization:
Let's first write 24 as a product of its prime factors.
[tex]\[
24 = 2^3 \times 3
\][/tex]
2. Applying the Cube Root:
We need to find the cube root of 24, which can be expressed as:
[tex]\[
24^{\frac{1}{3}} = (2^3 \times 3)^{\frac{1}{3}}
\][/tex]
3. Distributing the Cube Root:
Distributing the cube root over the product, we get:
[tex]\[
24^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} \times 3^{\frac{1}{3}}
\][/tex]
4. Simplifying:
The cube root of [tex]\(2^3\)[/tex] is [tex]\(2\)[/tex], because:
[tex]\[
(2^3)^{\frac{1}{3}} = 2
\][/tex]
Therefore, we have:
[tex]\[
24^{\frac{1}{3}} = 2 \times 3^{\frac{1}{3}}
\][/tex]
5. Match with the Given Options:
We now compare [tex]\( 2 \times 3^{\frac{1}{3}} \)[/tex] with the given options:
- [tex]\(2 \sqrt{3}\)[/tex]
- [tex]\(2 \sqrt[3]{3}\)[/tex]
- [tex]\(2 \sqrt{6}\)[/tex]
- [tex]\(2 \sqrt[3]{6}\)[/tex]
It's clear that:
[tex]\[
2 \times 3^{\frac{1}{3}} = 2 \sqrt[3]{3}
\][/tex]
Therefore, the expression equivalent to [tex]\( 24^{\frac{1}{3}} \)[/tex] is:
[tex]\[
2 \sqrt[3]{3}
\][/tex]
