Write each rational expression in lowest terms. See Example 2. 8k + 16 9k + 18

Verified step by step guidance

1. Factor the numerator $8k+16$ by finding the greatest common factor (GCF). The GCF of 8 and 16 is 8, so factor out 8: $8(k+2)$.
2. Factor the denominator $9k+18$ by finding the greatest common factor (GCF). The GCF of 9 and 18 is 9, so factor out 9: $9(k+2)$.
3. Write the rational expression with the factored numerator and denominator: $\frac{8(k+2)}{9(k+2)}$.
4. Identify and cancel the common factor $(k+2)$ from the numerator and the denominator.
5. The expression in lowest terms is $\frac{8}{9}$.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. To simplify these expressions, one must factor both the numerator and the denominator to identify common factors that can be canceled out. Understanding how to manipulate polynomials is essential for working with rational expressions.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is crucial for simplifying rational expressions, as it allows for the identification of common factors in the numerator and denominator. Techniques such as factoring out the greatest common factor (GCF) or using special products (like the difference of squares) are commonly employed.

Lowest Terms

A rational expression is said to be in lowest terms when the numerator and denominator have no common factors other than 1. To achieve this, one must fully factor both parts and cancel any common factors. This simplification is important for clarity and accuracy in mathematical expressions, ensuring that the expression is as simple as possible.