https://file.aiquickdraw.com/mathbot/file/1727856303716opuzv5i1.jpg

Let's break down and solve the exercises from the image you provided:

### a) 
You have the expression:

\[
\frac{7}{2} + \frac{3}{5} = \frac{7.5 + 2.3}{2.5} = \frac{41}{10}
\]

This is a fraction addition problem. Here's a step-by-step explanation:
1. First, find a common denominator for the two fractions, which is \(2 \times 5 = 10\).
2. Rewrite the fractions:
   \[
   \frac{7}{2} = \frac{35}{10}, \quad \frac{3}{5} = \frac{6}{10}
   \]
3. Add them:
   \[
   \frac{35}{10} + \frac{6}{10} = \frac{41}{10}
   \]

The final result is \( \frac{41}{10} \).

### d) 
The expression given is:
\[
\frac{1}{x^2} + \frac{2}{xy} + \frac{1}{y^2}
\]
To combine these fractions, you need a common denominator, which would be \( x^2 \cdot y^2 \). Let’s rewrite and combine them:
1. The first fraction: \(\frac{1}{x^2}\) becomes \(\frac{y^2}{x^2y^2}\).
2. The second fraction: \(\frac{2}{xy}\) becomes \(\frac{2y}{x^2y^2}\).
3. The third fraction: \(\frac{1}{y^2}\) becomes \(\frac{x^2}{x^2y^2}\).

Now, combine the fractions:
\[
\frac{y^2 + 2xy + x^2}{x^2 y^2}
\]

### Additional Fraction Problem:
The next part shows a fraction of fractions:
\[
\frac{\frac{1}{5}}{\frac{2}{1}} = \frac{1}{5} \cdot \frac{1}{2} = \frac{1}{10}
\]

This step shows the division of fractions, where dividing by a fraction is the same as multiplying by its reciprocal. 

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Let me know if you'd like further details or have any other questions!

### Here are five related questions:
1. How do you find the least common denominator for adding fractions?
2. Can you simplify complex fractions without finding a common denominator?
3. How does multiplying and dividing fractions differ conceptually?
4. How can you generalize the process for adding three fractions with different denominators?
5. How do operations with exponents affect the simplification of fractions like the ones in part (d)?

### Tip:
When adding or subtracting fractions, always look for the least common denominator to simplify the calculations.

Learn how to add and simplify fractions, including algebraic fractions like 7/2 + 3/5 and 1/x^2 + 2/xy + 1/y^2. This solution breaks down fraction addition, finding a common denominator, and simplifying complex fractions. Suitable for students in Grades 8-10.

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