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.

$\frac{5}{6 a}, \frac{7}{8 a^{5}}$

Key Concepts

Rational Expressions

A rational expression is a quotient of two polynomials. These expressions can include variables, and just as with numerical fractions, operations such as addition, subtraction, multiplication, and division are performed by treating the numerator and denominator as algebraic expressions.

Least Common Denominator (LCD)

The least common denominator is the smallest expression that contains all the factors of the denominators of a set of rational expressions. It is crucial when combining or comparing rational expressions, as it provides a common basis to rewrite them as equivalent expressions with a shared denominator.

Identifying the LCD in Algebraic Fractions

To find the LCD in an algebraic context, each denominator is factored completely into its prime numerical and algebraic components. The LCD is formed by taking each distinct factor raised to the highest power with which it appears in any of the denominators, ensuring that each original denominator divides the LCD evenly.

Creating Equivalent Rational Expressions

Rewriting a rational expression to have a new denominator involves multiplying both the numerator and the denominator by a factor that transforms the original denominator into the LCD. This process, which maintains equality by effectively multiplying by 1, results in an equivalent form that is easier to work with when performing further calculations such as addition or subtraction.

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