Simplify each complex fraction. See Examples 5 and 6. x y — + — y x—————— x y — - — y x

Verified step by step guidance

1. Identify the complex fraction: \( \frac{\frac{x}{y} + \frac{y}{x}}{\frac{x}{y} - \frac{y}{x}} \).
2. Find a common denominator for the fractions in the numerator and the denominator. The common denominator for both is \( xy \).
3. Rewrite the numerator: \( \frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy} \).
4. Rewrite the denominator: \( \frac{x}{y} - \frac{y}{x} = \frac{x^2 - y^2}{xy} \).
5. Simplify the complex fraction by dividing the numerator by the denominator: \( \frac{\frac{x^2 + y^2}{xy}}{\frac{x^2 - y^2}{xy}} = \frac{x^2 + y^2}{x^2 - y^2} \).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify complex fractions, one typically finds a common denominator for the fractions involved, allowing for easier addition, subtraction, or simplification of the overall expression.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When simplifying complex fractions, identifying the least common denominator (LCD) is crucial, as it allows for the combination of fractions into a single fraction, making calculations more straightforward.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, distributing, and combining like terms, which are essential skills for simplifying complex fractions and ensuring that the final expression is in its simplest form.