Multiply or divide, as indicated. See Example 3. ((2k + 8) / 6) ÷ ((3k + 12) / 2)

Verified step by step guidance

1. Rewrite the division of the two fractions as multiplication by the reciprocal. The original expression is \( \frac{2k + 8}{6} \div \frac{3k + 12}{2} \), which can be rewritten as \( \frac{2k + 8}{6} \times \frac{2}{3k + 12} \).
2. Factor the numerators and denominators where possible. For example, factor out the greatest common factor (GCF) from \(2k + 8\) and \(3k + 12\): \(2k + 8 = 2(k + 4)\) and \(3k + 12 = 3(k + 4)\).
3. Substitute the factored forms back into the expression: \( \frac{2(k + 4)}{6} \times \frac{2}{3(k + 4)} \).
4. Simplify the expression by canceling common factors in the numerator and denominator. For example, cancel \(k + 4\) and reduce numeric coefficients where possible.
5. Multiply the remaining numerators together and the denominators together to get the simplified product.

Key Concepts

Multiplication and Division of Fractions

Dividing fractions involves multiplying by the reciprocal of the divisor. For example, dividing by a fraction is the same as multiplying by its inverse. This principle allows complex fraction expressions to be simplified by converting division into multiplication.

Factoring Algebraic Expressions

Factoring involves rewriting expressions as products of simpler factors. Recognizing common factors, such as constants or variable terms, helps simplify expressions before performing operations like multiplication or division, making calculations more manageable.

Simplifying Rational Expressions

Simplifying rational expressions means reducing fractions by canceling common factors in the numerator and denominator. This process is essential after factoring to obtain the simplest form of the expression, which aids in clearer interpretation and further calculations.