Questions: Find the LCD of the rational expressions in the list. 8/(5x+15), 7/(6x-48) The LCD (least common denominator) is □ (Type your answer in factored form.)

Transcript text: Find the LCD of the rational expressions in the list. \[ \frac{8}{5 x+15}, \frac{7}{6 x-48} \] The LCD (least common denominator) is $\square$ (Type your answer in factored form.)

Solution

Solution Steps

To find the Least Common Denominator (LCD) of the given rational expressions, we need to factor the denominators and then determine the least common multiple (LCM) of these factors.

Factor the denominators $5x + 15$ and $6x - 48$.
Identify the unique factors from both denominators.
The LCD will be the product of the highest powers of all unique factors.

Step 1: Factor the Denominators

We start by factoring the denominators of the given rational expressions: \[ 5x + 15 = 5(x + 3) \] \[ 6x - 48 = 6(x - 8) \]

Step 2: Identify Unique Factors

The unique factors from the factored denominators are:

From $5(x + 3)$: $5$ and $(x + 3)$
From $6(x - 8)$: $6$ and $(x - 8)$

Step 3: Calculate the Least Common Denominator (LCD)

To find the LCD, we take the highest power of each unique factor: \[ \text{LCD} = 30(x + 3)(x - 8) \] This is derived from the product of the highest coefficients and the unique factors.