To solve the given problem, we need to perform the addition of two rational expressions and write the result in its simplest form.
The given rational expressions are:
[tex]\[
\frac{5}{5 n} + \frac{2}{6 m}
\][/tex]
### Step 1: Simplify the individual fractions
First, we simplify each fraction separately:
1. Simplify [tex]\(\frac{5}{5 n}\)[/tex]:
[tex]\[
\frac{5}{5 n} = \frac{5 \div 5}{5 n \div 5} = \frac{1}{n}
\][/tex]
2. Simplify [tex]\(\frac{2}{6 m}\)[/tex]:
[tex]\[
\frac{2}{6 m} = \frac{2 \div 2}{6 m \div 2} = \frac{1}{3 m}
\][/tex]
### Step 2: Find a common denominator
To add the fractions, we need a common denominator. The denominators are [tex]\(n\)[/tex] and [tex]\(3m\)[/tex]. The least common multiple (LCM) of [tex]\(n\)[/tex] and [tex]\(3m\)[/tex] is [tex]\(3mn\)[/tex].
### Step 3: Adjust the numerators to the common denominator
Convert each fraction to have the common denominator [tex]\(3mn\)[/tex].
1. Convert [tex]\(\frac{1}{n}\)[/tex]:
[tex]\[
\frac{1}{n} = \frac{1 \cdot 3m}{n \cdot 3m} = \frac{3m}{3mn}
\][/tex]
2. Convert [tex]\(\frac{1}{3 m}\)[/tex]:
[tex]\[
\frac{1}{3 m} = \frac{1 \cdot n}{3m \cdot n} = \frac{n}{3mn}
\][/tex]
### Step 4: Add the fractions
Now we can add the fractions as they have the same denominator:
[tex]\[
\frac{3m}{3mn} + \frac{n}{3mn} = \frac{3m + n}{3mn}
\][/tex]
### Step 5: Simplify if possible
The fraction [tex]\(\frac{3m + n}{3mn}\)[/tex] is already in its simplest form. No further simplification is needed.
### Conclusion
The simplified form of the given rational expressions [tex]\(\frac{5}{5 n} + \frac{2}{6 m}\)[/tex] is:
[tex]\[
\frac{3 m + n}{3 m n}
\][/tex]
Among the given options:
- [tex]\(\frac{7}{11 m n}\)[/tex]
- [tex]\(\frac{7}{5 n + 6 m}\)[/tex]
- [tex]\(\frac{3 m + n}{3 m n}\)[/tex]
- [tex]\(\frac{6 m + 2 n}{m n}\)[/tex]
The correct answer is:
[tex]\[
\frac{3 m + n}{3 m n}
\][/tex]
