Simplify each complex fraction. See Examples 5 and 6. (x/y + y/x) / (x/y − y/x)

Verified step by step guidance

1. Rewrite the complex fraction clearly as a division of two fractions: the numerator is \(\frac{x}{y} + \frac{y}{x}\) and the denominator is \(\frac{x}{y} - \frac{y}{x}\).
2. Find a common denominator for the fractions in both the numerator and denominator. For each, the common denominator is \(xy\), so rewrite each fraction accordingly: \(\frac{x}{y} = \frac{x^2}{xy}\) and \(\frac{y}{x} = \frac{y^2}{xy}\).
3. Combine the fractions in the numerator and denominator separately by adding or subtracting the numerators over the common denominator \(xy\): numerator becomes \(\frac{x^2 + y^2}{xy}\) and denominator becomes \(\frac{x^2 - y^2}{xy}\).
4. Rewrite the complex fraction as a division of two fractions: \(\frac{\frac{x^2 + y^2}{xy}}{\frac{x^2 - y^2}{xy}}\). Then, apply the rule for dividing fractions by multiplying the numerator by the reciprocal of the denominator.
5. Simplify the resulting expression by canceling the common denominator \(xy\) and express the final simplified form as \(\frac{x^2 + y^2}{x^2 - y^2}\).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying complex fractions involves rewriting them as a single simple fraction by finding common denominators or multiplying numerator and denominator by the least common denominator.

Fraction Addition and Subtraction

Adding or subtracting fractions requires a common denominator. Once denominators match, numerators are combined accordingly. This process is essential when simplifying expressions that involve sums or differences of fractions.

Algebraic Manipulation of Variables

When variables appear in fractions, it is important to treat them as algebraic expressions, applying rules of multiplication, division, and factoring carefully. Simplifying expressions with variables requires attention to domain restrictions and proper handling of terms.