Identify the least common denominator of each group of...

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator.

$\frac{7}{12 x-4}, \frac{x}{18 x-6}$

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. They are used to represent relationships between polynomial expressions and involve similar operations and restrictions as numerical fractions, especially regarding the domains where the denominators are not zero.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials or irreducible factors. This step is essential when working with rational expressions because it reveals the underlying factors in the denominators, which is necessary for finding a common denominator and simplifying the expressions.

Least Common Denominator (LCD)

The Least Common Denominator (LCD) is the smallest expression, derived from the factorized forms of individual denominators, that contains all the factors necessary to represent each denominator. It is used to rewrite rational expressions with different denominators into equivalent expressions with the same denominator, which is crucial for operations like addition or subtraction.

Equivalent Rational Expressions

Equivalent rational expressions are expressions that, while they may appear different, represent the same value. This equivalence is achieved by multiplying the numerator and the denominator by the same factor, often to express the original fraction with a new, common denominator, thus making it easier to combine or compare the expressions.

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