https://file.aiquickdraw.com/mathbot/file/1726846314785xdhar897.jpg

The expression shown in the image is:

\[
\frac{p+q}{p-q} + \frac{p-q}{p+q}
\]

Let's solve this step by step.

### Step 1: Find the common denominator
The common denominator between \(\frac{p+q}{p-q}\) and \(\frac{p-q}{p+q}\) is \((p+q)(p-q)\).

### Step 2: Rewrite the expression with the common denominator
\[
\frac{(p+q)^2 + (p-q)^2}{(p-q)(p+q)}
\]

### Step 3: Expand the numerator
\[
(p+q)^2 = p^2 + 2pq + q^2
\]
\[
(p-q)^2 = p^2 - 2pq + q^2
\]

So, the expression becomes:
\[
\frac{p^2 + 2pq + q^2 + p^2 - 2pq + q^2}{(p-q)(p+q)}
\]

### Step 4: Simplify the numerator
\[
p^2 + 2pq + q^2 + p^2 - 2pq + q^2 = 2p^2 + 2q^2
\]

Thus, the expression is now:
\[
\frac{2p^2 + 2q^2}{(p-q)(p+q)}
\]

### Step 5: Factor out the 2 in the numerator
\[
\frac{2(p^2 + q^2)}{(p-q)(p+q)}
\]

### Final Answer:
The simplified result is:
\[
\boxed{\frac{2(p^2 + q^2)}{(p-q)(p+q)}}
\]

Would you like more details on any part of this solution?

Here are 5 related questions you might find helpful:
1. How do you simplify expressions with common denominators?
2. What is the process for expanding binomials like \((p+q)^2\)?
3. Can you derive the difference of squares formula from \((p+q)(p-q)\)?
4. How can we generalize this process for similar expressions?
5. What happens if \(p = q\) in this expression?

**Tip**: When adding fractions, always look for a common denominator before attempting to simplify!

Learn how to simplify the algebraic expression (p+q)/(p-q) + (p-q)/(p+q) step by step. This solution covers algebraic concepts such as finding a common denominator, expanding binomials, and using the difference of squares formula. Ideal for students in Grades 9-11.

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