Simplify each complex fraction. [ 1/(x³-y³) ] / [ 1/(x² -y²) ]

Verified step by step guidance

1. Identify the complex fraction given: \(\frac{\frac{1}{x^3 - y^3}}{\frac{1}{x^2 - y^2}}\).
2. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. So rewrite the expression as \(\frac{1}{x^3 - y^3} \times \frac{x^2 - y^2}{1}\).
3. Factor the expressions in the numerator and denominator where possible. Use the difference of cubes formula for \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\) and the difference of squares formula for \(x^2 - y^2 = (x - y)(x + y)\).
4. Substitute the factored forms back into the expression: \(\frac{1}{(x - y)(x^2 + xy + y^2)} \times (x - y)(x + y)\).
5. Cancel the common factor \((x - y)\) from numerator and denominator, then write the simplified expression as \(\frac{x + y}{x^2 + xy + y^2}\).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the complex fraction as a division problem and then multiplying by the reciprocal of the denominator fraction to simplify the expression.

Difference of Squares and Cubes

The expressions x^2 - y^2 and x^3 - y^3 are special polynomial forms. x^2 - y^2 factors into (x - y)(x + y), while x^3 - y^3 factors into (x - y)(x^2 + xy + y^2). Recognizing these helps simplify expressions by canceling common factors.

Multiplying and Dividing Rational Expressions

When dividing rational expressions, multiply by the reciprocal of the divisor. Simplify by factoring numerators and denominators, then cancel common factors. This process reduces complex fractions to simpler forms.