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The given expression is:

\[
\frac{y - \frac{\frac{y^2}{y - x}}{1 + \frac{\frac{x^2}{y - x}}{1}}}
\]

Let's break it down and simplify step by step.

1. Start by simplifying the denominator inside the overall fraction:

   \[
   1 + \frac{\frac{x^2}{y - x}}{1} = 1 + \frac{x^2}{y - x}
   \]

2. Now, substitute this back into the original expression:

   \[
   y - \frac{\frac{y^2}{y - x}}{1 + \frac{x^2}{y - x}}
   \]

3. Combine the terms in the fraction:

   \[
   y - \frac{y^2}{(y - x) + x^2}
   \]

4. This is the simplified expression. Notice that it depends on the values of \( y \) and \( x \), and further factorization is not possible unless specific values are provided.

Would you like more details, or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would the expression change if \( x \) and \( y \) were equal?
2. What happens if \( y = 0 \) in the simplified expression?
3. Can the expression be further simplified if \( x = 1 \)?
4. How does substituting specific values for \( y \) and \( x \) affect the overall fraction?
5. Could the expression have any undefined points where the denominator becomes zero?

**Tip:** Always look for common factors in the numerator and denominator when simplifying complex fractions!

Learn how to simplify a complex fraction step-by-step. This problem involves fraction simplification, complex fractions, and variable substitution. Suitable for advanced high school students and those preparing for college-level mathematics.

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