To find the equivalent expression for [tex]\(\frac{4 f^2}{3} + \frac{1}{4 f}\)[/tex], let's go through a step-by-step simplification process using fraction algebra.
1. Start with the given expression:
[tex]\[
\frac{4 f^2}{3} + \frac{1}{4 f}
\][/tex]
2. To combine these two fractions, we need to find a common denominator. The denominators are 3 and [tex]\(4f\)[/tex]. The common denominator will be [tex]\(12f\)[/tex] (since [tex]\(12f\)[/tex] is the least common multiple of 3 and [tex]\(4f\)[/tex]).
3. Rewrite each fraction with [tex]\(12f\)[/tex] as the denominator:
[tex]\[
\frac{4 f^2}{3} = \frac{4 f^2 \cdot 4 f}{3 \cdot 4 f} = \frac{16 f^3}{12 f}
\][/tex]
[tex]\[
\frac{1}{4 f} = \frac{1 \cdot 3}{4 f \cdot 3} = \frac{3}{12 f}
\][/tex]
4. Now we can add the two fractions together, since they have a common denominator:
[tex]\[
\frac{16 f^3}{12 f} + \frac{3}{12 f} = \frac{16 f^3 + 3}{12 f}
\][/tex]
5. Therefore, the simplified form of the given expression is:
[tex]\[
\frac{16 f^3 + 3}{12 f}
\][/tex]
Now we can compare this simplified expression with the given multiple-choice options:
- [tex]\(\frac{16 t^3}{3}\)[/tex]
- [tex]\(\frac{f}{3}\)[/tex]
- [tex]\(\frac{3}{16 f^3}\)[/tex]
- [tex]\(\frac{3}{7}\)[/tex]
The correct match is [tex]\(\frac{16 f^3 + 3}{12 f}\)[/tex], which is not explicitly listed among the original options. This appears to be an error in the options provided, as none match the simplified expression exactly. Based on the simplification process, the equivalent expression is indeed:
[tex]\[
\frac{16 f^3 + 3}{12 f}
\][/tex]
