The factorizations of the denominators of two rational...

The factorizations of the denominators of two rational expressions are given. Find the LCD.
a. $12 a=2 \cdot 2 \cdot 3 \cdot a$
$18 a^{2}=2 \cdot 3 \cdot 3 \cdot a \cdot a$
b. $x^{2}-36=(x+6)(x-6)$
$3 x-18=3(x-6)$

Key Concepts

Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that contains all the factors present in the denominators of a set of rational expressions at their highest powers. It is used to combine fractions by ensuring that all denominators are expressed in a common form, making addition, subtraction, or comparison of rational expressions possible.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. They often necessitate the computation of a common denominator, particularly when performing operations such as addition and subtraction, making the concepts of factoring and LCD vital for their manipulation and simplification.

Prime Factorization

Prime factorization is the process of expressing a number or an algebraic expression as a product of its prime factors. This concept is essential when working with rational expressions because it allows you to break down denominators into their simplest components, facilitating the identification of common factors among different terms.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials or binomials. This technique is crucial for identifying factors of algebraic expressions in denominators, which in turn is necessary for determining the common denominator when combining rational expressions.

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