To determine which expression is equivalent to [tex]\(\sqrt[3]{5}\)[/tex], we first need to understand what [tex]\(\sqrt[3]{5}\)[/tex] or the cube root of 5 means. The cube root of a number [tex]\(a\)[/tex] is the number [tex]\(b\)[/tex] such that [tex]\(b^3 = a\)[/tex].
In mathematical terms, the cube root of a number is equivalent to raising that number to the power of [tex]\(\frac{1}{3}\)[/tex]. This is expressed as:
[tex]\[ \sqrt[3]{5} = 5^{\frac{1}{3}} \][/tex]
Now, let's go through each of the given options:
A. [tex]\(\frac{3}{5}\)[/tex] – This is the fraction three-fifths and does not represent the cube root of 5.
B. [tex]\(5^{\frac{1}{3}}\)[/tex] – This is the notation for raising 5 to the power of one-third, which is exactly equivalent to taking the cube root of 5.
C. [tex]\(\frac{5}{3}\)[/tex] – This is the fraction five-thirds and also does not represent the cube root of 5.
D. [tex]\(5^3\)[/tex] – This is 5 raised to the power of 3, which represents [tex]\(5 \times 5 \times 5\)[/tex], and it is the opposite operation of taking the cube root.
Thus, the correct expression that represents [tex]\(\sqrt[3]{5}\)[/tex] is option B, which is [tex]\(5^{\frac{1}{3}}\)[/tex].
Therefore, the correct answer is:
[tex]\[ B. \, 5^{\frac{1}{3}} \][/tex]
