Simplify each complex fraction. [ 1/(a³+b³) ] / [ 1/(a² + 2ab + b²) ]

Verified step by step guidance

1. Identify the complex fraction: \(\frac{\frac{1}{a^3 + b^3}}{\frac{1}{a^2 + 2ab + b^2}}\).
2. Recall that dividing by a fraction is the same as multiplying by its reciprocal, so rewrite the expression as \(\frac{1}{a^3 + b^3} \times \frac{a^2 + 2ab + b^2}{1}\).
3. Recognize and factor the denominators where possible: factor \(a^3 + b^3\) using the sum of cubes formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), and factor \(a^2 + 2ab + b^2\) as a perfect square trinomial \((a + b)^2\).
4. Substitute the factored forms back into the expression to get \(\frac{1}{(a + b)(a^2 - ab + b^2)} \times (a + b)^2\).
5. Simplify by canceling common factors: cancel one \((a + b)\) from numerator and denominator, leaving \(\frac{a + b}{a^2 - ab + b^2}\) as the simplified expression.

Key Concepts

Complex Fractions

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

Sum of Cubes Factorization

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). Recognizing and applying this factorization helps simplify expressions involving sums of cubes by breaking them into simpler polynomial factors.

Perfect Square Trinomials

A perfect square trinomial is an expression like a² + 2ab + b², which factors into (a + b)². Identifying this pattern allows for easier simplification and manipulation of algebraic expressions.