To simplify the given expression, let's break it down step by step. The expression is: \[ \frac{y - \frac{y^2}{y - x}}{1 + \frac{x^2}{(y - x)^2}} \] ### Step 1: Simplify the numerator The numerator is: \[ y - \frac{y^2}{y - x} \] We can combine the terms by getting a common denominator: \[ \frac{y(y - x) - y^2}{y - x} = \frac{y^2 - xy - y^2}{y - x} = \frac{-xy}{y - x} \] ### Step 2: Simplify the denominator The denominator is: \[ 1 + \frac{x^2}{(y - x)^2} \] This can be written as: \[ \frac{(y - x)^2 + x^2}{(y - x)^2} \] ### Step 3: Combine numerator and denominator Now the whole expression becomes: \[ \frac{\frac{-xy}{y - x}}{\frac{(y - x)^2 + x^2}{(y - x)^2}} = \frac{-xy \cdot (y - x)^2}{(y - x) \cdot ((y - x)^2 + x^2)} \] Canceling out one \((y - x)\) from the numerator and the denominator: \[ \frac{-xy \cdot (y - x)}{(y - x)^2 + x^2} \] ### Final Simplified Expression: The simplified expression is: \[ \frac{-xy(y - x)}{(y - x)^2 + x^2} \] This is the most simplified form of the given expression. Do you want more details, or do you have any other questions? Here are 5 related questions to expand your understanding: 1. What happens if \(y = x\) in the original expression? 2. How would the simplification change if the expression had different powers in the numerator and denominator? 3. Can the expression be simplified further if specific values for \(x\) and \(y\) are given? 4. How does this simplification change if additional terms are added to the numerator and denominator? 5. How would you approach the simplification if the expression were more complex with additional variables? **Tip:** Always check for opportunities to factor and cancel terms when simplifying algebraic expressions!