Question

which expression is equivalent to \\(\frac{4f^2}{3} \div \frac{1}{4f}\\)? \\(\frac{16f^3}{3}\\) \\(\frac{f}{3}\\) \\(\frac{3}{16f^3}\\) \\(\frac{3}{f}\\)

Explanation:

Step1: Find a common denominator

To add the two fractions \(\frac{4f^{2}}{3}\) and \(\frac{1}{4f}\), we need a common denominator. The least common denominator (LCD) of 3 and \(4f\) is \(12f\).

Step2: Rewrite each fraction with the LCD

For the first fraction \(\frac{4f^{2}}{3}\), multiply numerator and denominator by \(4f\):
\[
\frac{4f^{2}}{3} \times \frac{4f}{4f} = \frac{16f^{3}}{12f}
\]
For the second fraction \(\frac{1}{4f}\), multiply numerator and denominator by 3:
\[
\frac{1}{4f} \times \frac{3}{3} = \frac{3}{12f}
\]

Step3: Add the fractions

Now that both fractions have the same denominator, we can add them:
\[
\frac{16f^{3}}{12f} + \frac{3}{12f} = \frac{16f^{3} + 3}{12f}
\]
Wait, this seems to be a mistake. Wait, the original problem is \(\frac{4f^{2}}{3} \div \frac{1}{4f}\) (maybe a typo in the question, because the options don't match addition). Let's re - evaluate. If it's division: \(\frac{4f^{2}}{3}\div\frac{1}{4f}\)

Step1: Recall division of fractions rule

The rule for dividing fractions is \(a\div b=a\times\frac{1}{b}\) (where \(b
eq0\)). So \(\frac{4f^{2}}{3}\div\frac{1}{4f}=\frac{4f^{2}}{3}\times4f\)

Step2: Multiply the numerators and denominators

Multiply the coefficients and the variables separately. The coefficient part: \(4\times4 = 16\), the variable part: \(f^{2}\times f=f^{3}\), and the denominator is 3. So \(\frac{4f^{2}}{3}\times4f=\frac{16f^{3}}{3}\)

Answer:

\(\frac{16f^{3}}{3}\) (corresponding to the first option)