Let's start by carefully analyzing and simplifying the given expression step by step:
Given expression:
[tex]$\frac{4 f^2}{3} \div \frac{1}{4 f}$[/tex]
To simplify this, we can use the property of division of fractions which states:
[tex]$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$[/tex]
So, we should rewrite the division as a multiplication by the reciprocal of the second fraction:
[tex]$\frac{4 f^2}{3} \div \frac{1}{4 f} = \frac{4 f^2}{3} \times \frac{4 f}{1}$[/tex]
Now, we multiply the numerators and the denominators:
[tex]$\frac{4 f^2 \times 4 f}{3 \times 1} = \frac{4 \times 4 \times f^2 \times f}{3} = \frac{16 f^3}{3}$[/tex]
Thus, we have the simplified expression:
[tex]$\frac{16 f^3}{3}$[/tex]
Among the given options, the equivalent expression is:
[tex]$\frac{16 f^3}{3}$[/tex]
Therefore, the correct answer is:
[tex]$\frac{16 f^3}{3}$[/tex]
