Problem 10 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{3}{x-6} \text { and } \frac{4}{x^{2}-36}$$

Step-by-Step Solution

Answer

The least common denominator of the two rational expressions is \((x+6)(x-6)\).

Step 1: Identify the Denominators

Identify the denominators of the given fractions. They are \(x-6\) and \(x^{2}-36\).

Step 2: Factorize the Denominators

Factorize the denominators, if possible. The first one, \(x-6\), cannot be factorized further. However, the second denominator, \(x^{2}-36\), is a difference of two squares and can be factorized as \((x+6)(x-6)\).

Step 3: Find the Least Common Denominator

The least common denominator (LCD) is the expression that includes all factors of each denominator, in the greatest quantity they occur. Combining the factors from both fractions, the LCD is \((x+6)(x-6)\).