Simplify each complex fraction. See Examples 5 and 6. 4 - — 3———— 2 — 9

Verified step by step guidance

1. Identify the complex fraction: \( \frac{\frac{4}{3}}{\frac{2}{9}} \).
2. To simplify, multiply the numerator by the reciprocal of the denominator: \( \frac{4}{3} \times \frac{9}{2} \).
3. Multiply the numerators together: \( 4 \times 9 \).
4. Multiply the denominators together: \( 3 \times 2 \).
5. Simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor.

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 fractions is essential for effectively simplifying complex fractions.

Finding Common Denominators

Finding a common denominator is crucial when adding, subtracting, or simplifying fractions. The common denominator is a shared multiple of the denominators involved, allowing for the fractions to be combined or simplified. This concept is particularly important in complex fractions, where multiple layers of fractions must be unified to simplify the expression.

Simplification of Fractions

Simplification of fractions involves reducing them to their lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD). This process makes fractions easier to work with and understand. In the context of complex fractions, simplification is the final step after resolving the inner fractions and finding a common denominator.