Find the LCD of the rational expressions.

How to Find the LCD of Two Rational Expressions

• Answer:

  The LCD, short for least or lowest common denominator, is the smallest integer that is a multiple of a denominator. The main purpose of finding the LCD is to make the addition and subtraction of rational functions (fractions) with unlike denominators possible. The process for finding the LCD is the same for both integer fraction and rational expressions that contain variables in the denominator. In solving for the addition and subtraction operations between rational expressions with unlike denominators, finding the LCD involves finding a common denominator containing each differing component between the individual denominators. Difficulty: Moderately Easy Instructions Find the least common multiple between the denominators. To do this, set up a chart listing each denominator separately in columns and factor them. Find the common factor by observing each unique multiple in each column. For example, for the expression (5 / 2x) + (9x / x^2) + (2 / x) list the factors of each denominator: 2x: x * 2; x^2: x * x; and x: x. The first factors have an x in common, the second factors have only one x and the third factors have only a 2. Combining these terms finds: x * x * 2 = 2x^2. Determine what factor each rational expression must be multiplied by to obtain the common denominator. For example, for the rational expression (5 / 2x) + (9x / x^2) + (2 / x) with an LCD of 2x^2, it can be seen that (5 / 2x) must be multiplied by (x / x) to obtain the LCD in its denominator, (9x / x^2) requires (2 / 2) and (2 / x) requires (2x / 2x). This is like multiplying each term by 1 while also giving the terms a common denominator. Multiply each term by the required value to obtain terms with common denominators. For example, (5 / 2x) * (x / x) = (5x / 2x^2); (9x / x^2) * (2 / 2) = (18x / 2x^2); (2 / x) * (2x / 2x) = (4x / 2x^2). Combine the terms to obtain a rational expression over a common denominator. For example, (5 / 2x) + (9x / x^2) + (2 / x) = (5x / 2x^2) + (18x / 2x^2) + (4x / 2x^2). Solve the original expression in terms of its least common denominator. For example, (5x + 18x + 4x / 2x^2) = (27x / 2x^2) = (27 / 2x).