To determine which expression is equivalent to [tex]\(24^{\frac{1}{3}}\)[/tex], let's carefully work through the math step by step.
First, recall the property of exponents that allows us to rewrite numbers in terms of their prime factors. The number 24 can be factored as:
[tex]\[ 24 = 2^3 \times 3 \][/tex]
Next, let's take the cube root of 24. Using our prime factorization, we get:
[tex]\[ 24^{\frac{1}{3}} = (2^3 \times 3)^{\frac{1}{3}} \][/tex]
Now we apply the properties of exponents to separate the factors under the cube root:
[tex]\[ (2^3 \times 3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} \times 3^{\frac{1}{3}} \][/tex]
Since [tex]\( \frac{3}{3} = 1 \)[/tex], this simplifies to:
[tex]\[ 2^1 \times 3^{\frac{1}{3}} = 2 \times 3^{\frac{1}{3}} \][/tex]
Therefore, the expression [tex]\(24^{\frac{1}{3}}\)[/tex] is equivalent to:
[tex]\[ 2 \sqrt[3]{3} \][/tex]
So the correct answer is:
[tex]\[ 2 \sqrt[3]{3} \][/tex]
