To determine which expression is equivalent to [tex]\(24^{\frac{1}{3}}\)[/tex], we need to evaluate the expressions given in the options and compare the results to the value of [tex]\(24^{\frac{1}{3}}\)[/tex].
First, let's find the value of [tex]\(24^{\frac{1}{3}}\)[/tex]:
[tex]\[ 24^{\frac{1}{3}} \approx 2.8845 \][/tex]
Now, evaluate each given option:
1. [tex]\(2 \sqrt{3}\)[/tex]:
[tex]\[ 2 \sqrt{3} = 2 \times \sqrt{3} \approx 2 \times 1.7321 \approx 3.4641 \][/tex]
2. [tex]\(2 \sqrt[3]{3}\)[/tex]:
[tex]\[ 2 \sqrt[3]{3} = 2 \times 3^{\frac{1}{3}} \approx 2 \times 1.4422 \approx 2.8845 \][/tex]
3. [tex]\(2 \sqrt{6}\)[/tex]:
[tex]\[ 2 \sqrt{6} = 2 \times \sqrt{6} \approx 2 \times 2.4495 \approx 4.8990 \][/tex]
4. [tex]\(\sqrt[3]{6}\)[/tex]:
[tex]\[ \sqrt[3]{6} = 6^{\frac{1}{3}} \approx 1.8171 \][/tex]
Comparing these results, we see that:
- [tex]\(24^{\frac{1}{3}} \approx 2.8845\)[/tex]
- [tex]\(2 \sqrt{3} \approx 3.4641\)[/tex]
- [tex]\(2 \sqrt[3]{3} \approx 2.8845\)[/tex]
- [tex]\(2 \sqrt{6} \approx 4.8990\)[/tex]
- [tex]\(\sqrt[3]{6} \approx 1.8171\)[/tex]
Thus, the expression [tex]\(2 \sqrt[3]{3}\)[/tex] matches [tex]\(24^{\frac{1}{3}}\)[/tex].
So, the expression equivalent to [tex]\(24^{\frac{1}{3}}\)[/tex] is:
[tex]\[ \boxed{2 \sqrt[3]{3}} \][/tex]
