https://file.aiquickdraw.com/mathbot/file/1727331841311och10wpd.jpg

The expression you provided is:

\[
\frac{5}{7x} + \frac{6}{11x^5}
\]

To add these fractions, we need to find a common denominator. The denominators are \(7x\) and \(11x^5\).

### Step 1: Find the least common denominator (LCD)
The LCD will be the product of the two denominators since they share the common factor \(x\), but with different powers:
\[
\text{LCD} = 77x^5
\]

### Step 2: Rewrite each fraction with the LCD as the denominator
To rewrite each fraction:

- For \(\frac{5}{7x}\), multiply the numerator and denominator by \(x^4\) to get:
\[
\frac{5}{7x} = \frac{5x^4}{77x^5}
\]

- For \(\frac{6}{11x^5}\), multiply the numerator and denominator by 7 to get:
\[
\frac{6}{11x^5} = \frac{42}{77x^5}
\]

### Step 3: Add the fractions
Now that both fractions have the same denominator, we can add them directly:
\[
\frac{5x^4}{77x^5} + \frac{42}{77x^5} = \frac{5x^4 + 42}{77x^5}
\]

### Final Answer:
\[
\frac{5x^4 + 42}{77x^5}
\]

Let me know if you'd like more details or further assistance.

Here are 5 related questions to explore:
1. How would you simplify the expression if one of the powers of \(x\) was different?
2. Can you combine the terms if the exponents on \(x\) were the same?
3. How do you determine the least common denominator when combining rational expressions?
4. What is the next step if this expression were part of an equation to solve for \(x\)?
5. How does the process change when combining three or more rational expressions?

**Tip:** Always factor the denominators when combining fractions, as it makes finding the least common denominator easier.

This guide explains how to add fractions with different denominators using algebra, specifically focusing on finding the least common denominator and simplifying the resulting fraction. Ideal for students in grades 8 to 10 learning algebraic fraction operations.

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