Simplify each complex fraction. See Examples 5 and 6. (−4/3) ÷ (2/9)

Verified step by step guidance

1. Rewrite the complex fraction clearly as \(\frac{\frac{4}{3}}{\frac{2}{9}}\) to understand the structure better.
2. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. So, rewrite the expression as \(\frac{4}{3} \times \frac{9}{2}\).
3. Multiply the numerators together and the denominators together: numerator = \(4 \times 9\), denominator = \(3 \times 2\).
4. Simplify the resulting fraction by performing the multiplications and then reducing the fraction to its simplest form by dividing numerator and denominator by their greatest common divisor.
5. Express the simplified fraction as the final answer, ensuring it is in lowest terms.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression so that it no longer contains fractions within fractions, often by finding a common denominator or multiplying numerator and denominator by the least common denominator.

Reciprocal and Division of Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. To simplify complex fractions, you often convert division into multiplication by flipping the denominator fraction, which makes the expression easier to handle and simplify.

Simplifying Fractions

After rewriting the complex fraction, simplify by reducing fractions to their lowest terms. This involves factoring numerators and denominators, canceling common factors, and performing arithmetic operations to achieve the simplest form.