Simplify each complex fraction. See Examples 5 and 6. ( -4⁄3 ) + 12⁄5———————— 1 - ( -4⁄3 ) (12⁄5)

Verified step by step guidance

1. Step 1: Simplify the numerator: \( \left( -\frac{4}{3} \right) + \frac{12}{5} \). To do this, find a common denominator for the fractions.
2. Step 2: Simplify the denominator: \( 1 - \left( -\frac{4}{3} \right) \left( \frac{12}{5} \right) \). First, multiply the fractions \( -\frac{4}{3} \) and \( \frac{12}{5} \).
3. Step 3: Substitute the simplified expressions from Steps 1 and 2 back into the complex fraction.
4. Step 4: Simplify the complex fraction by dividing the simplified numerator by the simplified denominator.
5. Step 5: Simplify the resulting expression, if possible, to get the final simplified form.

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 crucial for effectively simplifying these types of expressions.

Operations with Fractions

Performing operations with fractions involves addition, subtraction, multiplication, and division. When adding or subtracting fractions, a common denominator is required, while multiplication involves multiplying the numerators and denominators directly. Mastery of these operations is essential for simplifying complex fractions accurately.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing for the multiplication of a single term by each term within a parenthesis. This property is often used in algebraic expressions to simplify calculations, especially when dealing with fractions and complex expressions. Recognizing when to apply this property is key to simplifying complex fractions effectively.