Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ]

Verified step by step guidance

1. Identify the complex fraction: \( \frac{\frac{1}{x^3-y^3}}{\frac{1}{x^2-y^2}} \).
2. Recall that dividing by a fraction is the same as multiplying by its reciprocal. Rewrite the expression as: \( \frac{1}{x^3-y^3} \times \frac{x^2-y^2}{1} \).
3. Multiply the numerators and the denominators: \( \frac{x^2-y^2}{x^3-y^3} \).
4. Factor the numerator \( x^2-y^2 \) using the difference of squares: \( (x-y)(x+y) \).
5. Factor the denominator \( x^3-y^3 \) using the difference of cubes: \( (x-y)(x^2+xy+y^2) \).

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 inner fractions and then simplifies the overall expression. Understanding how to manipulate these nested fractions is crucial for effective simplification.

Difference of Squares

The difference of squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). This concept is essential when simplifying expressions like x² - y², as it allows us to factor the expression into linear factors, making it easier to work with in complex fractions.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is a fundamental skill in algebra, as it simplifies expressions and helps in solving equations. In the context of the given problem, recognizing how to factor x³ - y³ and x² - y² will facilitate the simplification of the complex fraction.