To simplify the rational expression [tex]\(\frac{24 m^8 n^7 + 20 m^3 n^4 - 4 m n}{12 m^2 n^5}\)[/tex], let's break it down into individual steps:
1. Write down the original expression:
[tex]\[
\frac{24 m^8 n^7 + 20 m^3 n^4 - 4 m n}{12 m^2 n^5}
\][/tex]
2. Separate the numerator into separate fractions over the common denominator:
[tex]\[
\frac{24 m^8 n^7}{12 m^2 n^5} + \frac{20 m^3 n^4}{12 m^2 n^5} - \frac{4 m n}{12 m^2 n^5}
\][/tex]
3. Simplify each term individually:
- For the first term:
[tex]\[
\frac{24 m^8 n^7}{12 m^2 n^5} = \frac{24}{12} \cdot \frac{m^8}{m^2} \cdot \frac{n^7}{n^5} = 2 \cdot m^{8-2} \cdot n^{7-5} = 2 m^6 n^2
\][/tex]
- For the second term:
[tex]\[
\frac{20 m^3 n^4}{12 m^2 n^5} = \frac{20}{12} \cdot \frac{m^3}{m^2} \cdot \frac{n^4}{n^5} = \frac{5}{3} \cdot m^{3-2} \cdot n^{4-5} = \frac{5}{3} \cdot m \cdot n^{-1} = \frac{5m}{3n}
\][/tex]
- For the third term:
[tex]\[
\frac{4 m n}{12 m^2 n^5} = \frac{4}{12} \cdot \frac{m}{m^2} \cdot \frac{n}{n^5} = \frac{1}{3} \cdot m^{1-2} \cdot n^{1-5} = \frac{1}{3} \cdot m^{-1} \cdot n^{-4} = \frac{1}{3 m n^4}
\][/tex]
4. Combine all the simplified terms:
[tex]\[
2 m^6 n^2 + \frac{5 m}{3 n} - \frac{1}{3 m n^4}
\][/tex]
So, the simplified expression is:
[tex]\[
2 m^6 n^2 + \frac{5 m}{3 n} - \frac{1}{3 m n^4}
\][/tex]
This corresponds to option:
[tex]\[
2 m^6 n^2 + \frac{5 m}{3 n} - \frac{1}{3 m n^4}
\][/tex]
